that satisfies adjoint constraints is $f^{abc}$ – we cannot, for example,
modify a $d^{abc}$ color-structure with momentum-invariants to give it the
necessary antisymmetry properties (as is our method for generating adjoint
$c^{\text{HD}}$ weights from non-adjoint color at four points and beyond). As
such, we need only consider the following ordered cut:

$\displaystyle\lim_{s_{12}\to 0}s_{12}A_{5}^{\text{BC+HD}}({1,2,3,4,5})$
$\displaystyle=\sum_{s}A^{\rm bi-
adj}(1,2,l^{s})A^{\text{BC+HD}}(-l^{\bar{s}},3,4,5)$ (154)
$\displaystyle=\sum_{b}f^{a_{1}a_{2}b}A_{4}^{\text{BC+HD}}(-l^{b},3,4,5)\,$
(155)

where, since we are considering scalar amplitudes, the generic sum over states
reduces simply to a sum over the adjoint-color-index $b$ of the cut leg.

Given the local four-point higher-derivative corrections reviewed in Section
3, we must land on four-point amplitudes $A_{4}^{\text{BC+HD}}$ involving the
color structures $f^{3}f^{3}$ and $d^{4}$. In order for color to factorize
consistently, our factorizing five point amplitudes are thus restricted to
$c^{\text{HD}}$ involving color structures $f^{3}f^{3}f^{3}$ and $d^{4}f^{3}$.
Correspondingly, we will consider compositions of kinematics with the general
five point adjoint color numerators $c_{\mathfrak{a}}$ in eqs. (121) and
(122), given in terms of $f^{3}f^{3}f^{3}$ and $d^{4}f^{3}$ type color weights
respectively. The necessary spanning five-point candidate $c^{\text{HD}}$
numerators at a given mass dimension are then expressed simply through
repeated composition of each $c_{\mathfrak{a}}$ weight with an appropriate
number of linear relaxed numerator $\mathfrak{r}^{(1)}$ dressings as well as
products of these two $c_{\mathfrak{a}}$ weights with appropriate mass-
dimension scalar permutation invariants. We have verified explicitly that this
approach is sufficient to capture all factorizing solutions through mass-
dimension 18, or nine orders in momentum invariants.

Let us introduce the following notation for the nested application of
composition of $\mathfrak{r}^{(1)}$ with generic color weight $c$ to yield
higher-derivative adjoint weights:

$\left({\mathfrak{a}_{\mathfrak{r}}}^{({n})}({c})\right)=\left\\{\begin{array}[]{ll}\left(\mathfrak{r}^{(1)}\,$\mathfrak{a}$⃝\,\left({\mathfrak{a}_{\mathfrak{r}}}^{({n-1})}({c})\right)\right)&\quad
n>1\\\ \left(\mathfrak{r}^{(1)}\,$\mathfrak{a}$⃝\,c\right)&\quad
n=1\end{array}\right.$ (156)

(When $c=c_{\mathfrak{a}}$, we may also define
$\left({\mathfrak{a}_{\mathfrak{r}}}^{({0})}({c_{\mathfrak{a}}})\right)=c_{\mathfrak{a}}$;
for non-adjoint $c$, no such $n=0$ adjoint weight exists.) We find that such
continued nested compositions are needed until seventh order. Higher orders
hold no new adjoint structure modulo permutation invariants: we need only
consider products of lower order nested compositions with scalar permutation
invariants (themselves given ultimately solely by $\mathfrak{r}^{(1)}$, as
described earlier),

$\sum_{i+j=m}\,\left({\mathfrak{a}_{\mathfrak{r}}}^{({i})}({c_{\mathfrak{a}}})\right)\,\mathfrak{p}^{[j]}\quad\quad\quad\quad
m>7\,,$ (157)

in order to write down a structure that may be fixed on the cut to be
consistent with lower-point results. The specific building blocks necessary to
construct each order are summarized in Table 5. For clarity, we emphasize that
these building blocks are not themselves consistently factorizing solutions;
rather, they contain sufficient freedom to be fixed on eq. (154) such that
they factorize appropriately. The final number of fixed factorizing solutions
given in Table 5 at each order necessarily lines up with the number of Wilson
coefficients accompanying solutions of each color structure $d^{4}$ and
$f^{3}f^{3}$ in the four-point higher-derivative corrections
$\mathcal{A}_{4}^{\text{BC+HD}}$, as generated from the four-point modified
color factors $c^{\text{HD}}$ as given in eq. (56).

These nested compositions and their products with permutation invariants yield
all the expressions necessary for writing down factorizing amplitudes, as we
have verified through mass-dimension eighteen. We conjecture that this closed
structure above seventh order will persist, requiring only a finite number of
building blocks — the two adjoint color solutions and nested compositions and
permutation invariants, which all arise purely from $\mathfrak{r}^{(1)}$.

Table 5: Necessary building blocks to construct consistently factorizing adjoint modified color numerators $c^{\text{HD}}$, along with the final number of fixed factorizing solutions at each order. Order | Building Blocks |  $d^{4}f^{3}$ Solns |  $f^{3}f^{3}f^{3}$ Solns
---|---|---|---
2 |  $\left({\mathfrak{a}_{\mathfrak{r}}}^{({2})}({c_{\mathfrak{a}}})\right)$, $\mathfrak{p}^{(2)}\,c_{\mathfrak{a}}$ | 1 | 0
3 |  $\left({\mathfrak{a}_{\mathfrak{r}}}^{({3})}({c_{\mathfrak{a}}})\right)$, $\mathfrak{p}^{(3)}\,c_{\mathfrak{a}}$ | 0 | 1
4 | $\left({\mathfrak{a}_{\mathfrak{r}}}^{({4})}({c_{\mathfrak{a}}})\right)$ | 1 | 1
5 | $\left({\mathfrak{a}_{\mathfrak{r}}}^{({5})}({c_{\mathfrak{a}}})\right)$ | 1 | 1
6 | $\left({\mathfrak{a}_{\mathfrak{r}}}^{({6})}({c_{\mathfrak{a}}})\right)$ | 1 | 2
7 | $\left({\mathfrak{a}_{\mathfrak{r}}}^{({7})}({c_{\mathfrak{a}}})\right)$ | 1 | 2
8 | $\sum_{i+j=8}\left({\mathfrak{a}_{\mathfrak{r}}}^{({i})}({c_{\mathfrak{a}}})\right)\,\mathfrak{p}^{[j]}$ | 2 | 2
9 | $\sum_{i+j=9}\left({\mathfrak{a}_{\mathfrak{r}}}^{({i})}({c_{\mathfrak{a}}})\right)\,\mathfrak{p}^{[j]}$ | 1 | 3

###  6.2 Local contact higher-derivative corrections

Amplitudes arising from local five-point operators have no poles. At four
points, each cubic graph is dressed with only a single propagator, meaning
that only one cubic graph contributes to the residue on any given pole. Thus,
for all residues to vanish (as is required for a local contact amplitude), the
individual graph contributions had to be free of poles. At five points, there
are two propagators per cubic graph, so multiple graph channels contribute to
the same residue. As a result, we need not require that each channel be
individually local, but rather that the numerators conspire so that any
residues from one graph cancel with residues from other graphs across all
channels. One can identify local contact amplitudes by demanding that all cuts
of the functional ordered amplitude of the bicolored scalar theory vanish:

$\lim_{s_{i,i+1}\to
0}s_{i,i+1}A^{\text{BC+HD}}_{5}(12345)=0\quad\quad\quad\quad
i\in\\{1,2,3,4,5\\}$ (158)

where it is understood that $s_{5,6}\equiv s_{5,1}$. These conditions ensure
there are no residues on any of the poles of $A(12345)$. This is necessarily a
subset of higher-derivative adjoint color-weights. Such local weights are
needed to describe higher-derivative operators that contain no cubic or
quartic interactions, but only five-point and higher vertices, as such
operators cannot contribute to any amplitudes that factorize down to products
of four-point and three-point corrections.

Unlike factorizing amplitudes, local five-point contact amplitudes are not
restricted to any specific color structures, so we generically expect all
available color structures – $d^{5}$, $d^{4}f^{3}$, $f^{3}f^{3}f^{3}$, and
$f^{3}d^{3}f^{3}$ – to appear in local higher-derivative contact amplitudes.

The lowest order five-point contact amplitudes are found at fifth order in
momentum invariants (tenth order in mass-dimension) and contain the color
structures $d^{4}f^{3}$ and $f^{3}f^{3}f^{3}$. Only at sixth order does
$f^{3}d^{3}f^{3}$ color becomes relevant; $d^{5}$ color does not emerge until
ninth order. We have found that, through seventh order, nested composition
structures like those used for factorizing solutions, appearing in combination
with products with permutation invariants,

$\left({\mathfrak{a}_{\mathfrak{r}}}^{({m})}({c_{\mathfrak{a}}})\right)+\sum_{i+j=m}\left({\mathfrak{a}_{\mathfrak{r}}}^{({i})}({c_{\mathfrak{a}}})\right)p^{[j]}$
(159)

may be fixed on eq. (158) to give all $d^{4}f^{3}$ and $f^{3}f^{3}f^{3}$ local
contact amplitudes, as verified against an explicit ansatz calculation. At
sixth order, the $f^{3}d^{3}f^{3}$ local solution may be fixed from a nested
composition structure starting with sandwich color
$c_{\mathfrak{s}}=f^{3}d^{3}f^{3}$ rather than adjoint color
$c_{\mathfrak{a}}$:

$\left({\mathfrak{a}_{\mathfrak{r}}}^{({m})}({c_{\mathfrak{s}}})\right)+\sum_{i+j=m}\left({\mathfrak{a}_{\mathfrak{r}}}^{({i})}({c_{\mathfrak{s}}})\right)p^{[j]}$
(160)

Considering the size of the common denominator for the five-point cubic
graphs, imposing the condition of vanishing residues analytically quickly
becomes inefficient in comparison to a new method for generating local contact
amplitudes that we will describe in the next section. By uncovering novel
forms of color-kinematics duality and sharpening our focus on the local
contact condition, we will be able to climb several orders higher in mass-
dimension to find additional higher-derivative contact amplitudes.

##  7 Local Dualities Complementing Adjoint

At five points, we have identified a number of algebraic structures which span
single-trace color, and correspondingly many different color factors ($d^{5}$,
$d^{4}f^{3}$, $f^{3}f^{3}f^{3}$, or $f^{3}d^{3}f^{3}$). For generic higher-
derivative corrections, we must allow for all of these color structures to
appear in our amplitudes (corresponding to the most generic traces of
operators that may be written down in the action).

We find that we can use these new algebraic structures to our advantage by
considering simultaneous color-dual striations of the same full color-dressed
bi-color amplitudes. Specifically, we will introduce here an efficient method
for calculating five-point higher-derivative local counterterm predictions
that exploit two forms of color-kinematics duality using a manifestly local
construction.

###  7.1 Doubly-dual local amplitudes

All of our higher-derivative bicolored scalar amplitudes have a manifestly
adjoint double-copy structure involving two adjoint-type graph weights,

$\mathcal{A}^{\text{bi-
color}}=\sum_{g\in\Gamma_{3}}\frac{c^{\text{HD}}_{g}\tilde{c}_{g}}{d_{g}}\,.$
(161)

We encode higher-derivative corrections in the adjoint modified color factor
$c^{\text{HD}}$ through a mixture of kinematics and color-weights. The color-
weights themselves need not be adjoint; rather, the adjoint behavior of
$c^{\text{HD}}$ is ensured by kinematics acting in the right way to complement
the color-weights so that $c^{\text{HD}}$ obeys adjoint-type relations. All
higher-derivative corrections that can be associated with color-weights to
form adjoint-type modified color-factors are captured within $c^{\text{HD}}$.
We will generically refer to the type of color factors appearing in
$c^{\text{HD}}$ as $\mathfrak{c}$, where $\mathfrak{c}$ may be any of $d^{5}$,
$d^{4}f^{3}$, $f^{3}f^{3}f^{3}$, or $f^{3}d^{3}f^{3}$.

For local counterterm predictions, the scalar kinematics within
$c^{\text{HD}}_{g}$ conspire between graphs to cancel out the residue on all
possible poles of the amplitude, thus describing a five-point contact
interaction. At four points, contact interactions were described by simply
demanding that $c^{\text{HD}}_{g}$ be proportional to the unique propagator
associated with graph $g$; at five points, however, this cancellation of
denominators for cubic-graphs is non-trivial, as multiple graphs contribute to
each pole.

Next, we have the ubiquitous secondary adjoint color factor
$\tilde{c}=\tilde{f}^{3}\tilde{f}^{3}\tilde{f}^{3}$ of some additional gauge
group. This constitutes a simple adjoint stand-in that may be replaced in the
double copy with an adjoint vector-kinematic-weight like super Yang-Mills,
$\tilde{c}\rightarrow n^{\text{sYM}}$, to build higher-derivative corrections
to gauge theory. These adjoint color factors $\tilde{c}$ naturally provide an
adjoint-double-copy striation of the amplitude along cubic graphs, even though
the generic color factors $\mathfrak{c}$ living within $c^{\text{HD}}$ may not
traditionally lend themselves to a cubic graph description.

An interesting opportunity arises for $\mathfrak{c}$ whose natural structure
does not lie along cubic graphs, like $d^{4}f^{3}$ and $d^{5}$. We have the
freedom to choose to prioritize the $\mathfrak{c}$ color structures, using
them to provide an alternative striation of the amplitude along the graphs
$\Gamma_{\mathfrak{c}}$ relevant to their algebraic structure. After all, it
is the color structure $\mathfrak{c}$ that survives the double copy and
appears in the final gauge theory amplitude, rather than the adjoint color
$\tilde{c}=\tilde{f}^{3}\tilde{f}^{3}\tilde{f}^{3}$. Generically, bi-color
amplitudes can be written equivalently as a sum over the graphs relevant to
either structure, so we can choose to rewrite $\mathcal{A}^{\text{bi-color}}$
in terms of the graphs $\Gamma_{\mathfrak{c}}$ relevant to the color structure
$\mathfrak{c}$ inside the adjoint-type $c^{\text{HD}}$. Each of these graphs
is dressed with an appropriate color factor $\mathfrak{c}$ and a local
function $\mathfrak{f}_{\mathfrak{c}}$ of both the higher-derivative
kinematics that once resided in $c^{\text{HD}}$ and the remaining color
$\tilde{c}$,

$\mathcal{A}^{\text{bi-
color}}=\sum_{g\in\Gamma_{\mathfrak{c}}}\mathfrak{c}_{g}~{}\mathfrak{f}_{\mathfrak{c}}{}_{,g}\\!\left(s_{ij},\tilde{c}\right)$
(162)

The functions $\mathfrak{f}_{\mathfrak{c}}(g)$ can be fixed to two remarkable
properties: they are manifestly local, containing no propagator structure to
be cancelled non-trivially between graphs, and they are color-dual to the
relevant color structures $\mathfrak{c}$, meaning they obey the same algebraic
constraints as $\mathfrak{c}$-type color. To be clear, the functions of
$\mathfrak{f}_{\mathfrak{c}}$ need not be
$\mathfrak{f}_{\mathfrak{c}}$-ordered amplitudes, as we allow a summation over
all graphs $\Gamma_{\mathfrak{c}}$, not just the basis-graphs for the given
algebraic structure. As such, they are rather simply functions that dress
individual graphs, are manifestly local, and satisfy $\mathfrak{c}$-type
algebraic relations.

For example, for $\mathfrak{c}=d^{4}f^{3}$, the
$\mathfrak{f}_{\mathfrak{c}}(g)$ functions obey hybrid algebraic constraints
(symmetry in the first three arguments, antisymmetry in the last two, and
four-term Jacobi-like identities). The sum runs over the ten graphs that
contain one quartic vertex and one cubic vertex. For $\mathfrak{c}=d^{5}$,
there is just one graph (a single quintic vertex), and its
$\mathfrak{f}_{\mathfrak{c}}$ function obeys permutation invariant constraints
— a remarkably simpler description than demanding fifteen graph dressings
conspiring to cancel two propagators per graph!

Since these amplitudes enjoy two distinct color-dual striations, we refer to
them as doubly-dual local amplitudes, each admitting two equivalent
descriptions:

$\mathcal{A}^{\text{bi-
color}}=\sum_{g\in\Gamma_{3}}\frac{\tilde{c}_{g}~{}c^{\text{HD}}_{g}\\!\left(s_{ij},\mathfrak{c}\right)}{d_{g}}=\sum_{g\in\Gamma_{\mathfrak{c}}}\mathfrak{c}_{g}~{}\mathfrak{f}_{\mathfrak{c}}{}_{,g}\\!\left(s_{ij},\tilde{c}\right)$
(163)

The adjoint nature of the $\tilde{c}$-striated amplitude can still be
uncovered within the explicitly local $\mathfrak{c}$-dual striation: if the
$\tilde{c}$ color factors are expressed in a basis of traces, the coefficient
of each trace is an ordered amplitude that obeys the $(m-3)!=2$-basis BCJ
5-point amplitude relations required of any amplitude that may be written as
an adjoint double copy.

###  7.2 Composing to doubly-dual solutions

Multiple paths exist towards finding an appropriately doubly-dual local
$\mathfrak{f}_{\mathfrak{c}}(g)=\mathfrak{f}_{\mathfrak{c}}(g)\left(s_{ij},\tilde{c}\right)$
function. While it is worth mentioning that an ansatz approach to generating
$\mathfrak{f}_{\mathfrak{c}}(g)$ is more efficient than constructing an ansatz
for $c^{\text{HD}}$ (the $c^{\text{HD}}$ ansatz is larger, as it must
compensate for two propagators in the denominator, and, after fixing on
adjoint color-kinematics, must additionally be constrained to ensure all
residues vanish), both methods are surpassed by composition directly to dual
$\mathfrak{f}_{\mathfrak{c}}$ functions for some of our favorite algebraic
structures $\mathfrak{c}$, as composition ensures the
$\mathfrak{f}_{\mathfrak{c}}$ will obey the desired $\mathfrak{c}$-type
algebraic properties. Numerous $\mathfrak{c}$-dual functions can be formed at
each mass-dimension, but only particular combinations of them admit the type
of adjoint-double-copy structure we also require when striating the amplitude
along the $\tilde{c}$ color. We arrive at these particular combinations by
requiring that the adjoint $\tilde{c}$-striated ordered amplitudes satisfy the
adjoint $2$-basis amplitude relations. A summary of the total number of
doubly-dual solutions at each order in correction for the different color-
algebraic structures $\mathfrak{c}$ may be found in Table 6.

The existence of these two distinct color-kinematics dualities at play within
a single amplitude thus constitutes an efficient tool for generating local
solutions. Constructing the $\mathfrak{c}$-dual form first via composition is
computationally advantageous since it is manifestly local; once a
$\mathfrak{c}$-dual expression is found, its $\tilde{c}$-ordered amplitudes
may subsequently be fixed on the $2$-basis BCJ relations to impose adjoint
color-kinematics duality. To be explicit, the $\mathfrak{c}$-dual amplitude is
re-striated along the adjoint color $\tilde{c}$. Each coefficient of
$\tilde{c}$, when all $\tilde{c}$ have been expressed in a in a DDM basis,
must represent an adjoint-ordered amplitude if this expression is to be doubly
dual,

$\mathcal{A}=\sum_{g\in\Gamma_{\mathfrak{c}}}\mathfrak{c}_{g}~{}\mathfrak{f}_{\mathfrak{c}}{}_{,g}\\!\left(s_{ij},\tilde{c}\right)=\sum_{\sigma\in
S_{3}}\tilde{c}(1\sigma 5)A_{\mathfrak{c}}^{\text{HD}}(1\sigma 5)\,.$ (164)

Perhaps more familiar to many, and exactly identical, one can in turn express
the $\tilde{c}$ in terms of a trace basis, and look at each trace’s
coefficient. The coefficient of each independent trace must represent an
adjoint-ordered amplitude $A_{\mathfrak{c}}^{\text{HD}}$ for our expression to
be doubly dual:

$\mathcal{A}=\sum_{\sigma\in
S_{4}}\textrm{Tr}(\widetilde{T}^{a_{1}}\widetilde{T}^{a_{\sigma_{1}}}\cdots\widetilde{T}^{a_{\sigma_{4}}})A_{\mathfrak{c}}^{\text{HD}}(1\sigma)\,.$
(165)

These ordered amplitudes contain the scalar kinematics and $\mathfrak{c}$-type
color. We then require that these ordered amplitudes
$A_{\mathfrak{c}}^{\text{HD}}$ satisfy the $2$-basis BCJ relation: i

$s_{13}A_{\mathfrak{c}}^{\text{HD}}(13245)-s_{35}A_{\mathfrak{c}}^{\text{HD}}(12435)-(s_{34}+s_{35})A_{\mathfrak{c}}^{\text{HD}}(12345)=0\,.$
(166)

This is sufficient to ensure that all $2$-basis BCJ relations are satisfied
and that the full amplitude $\mathcal{A}$ has an adjoint color-dual
representation. Why? After establishing the $\mathfrak{c}$-dual algebra of our
$\mathfrak{f}_{\mathfrak{c}}(g)$ building blocks, our color-dressed
$\mathcal{A}$ is manifestly permutation invariant. This means that any
$A_{\mathfrak{c}}$ coefficient in a color basis either $\tilde{c}$ or it’s
associated trace basis will be related functionally by relabeling. It is
sufficient then to simply impose a single so-called fundamental BCJ relation.
The compatibility of these ordered amplitudes with the $2$-basis field theory
relations will eventually ensure that all doubly copy substitutions
$\tilde{c}\rightarrow n^{\text{vec}}_{\mathfrak{a}}$ yield fully gauge-
invariant vector amplitudes.

Table 6: Doubly dual local solutions. Order |  $d^{4}f^{3}$ Solutions |  $f^{3}f^{3}f^{3}$ Solutions |  $f^{3}d^{3}f^{3}$ Solutions |  $d^{5}$ Solutions | Total
---|---|---|---|---|---
5 | 1 | 1 | 0 | 0 | 2
6 | 2 | 2 | 1 | 0 | 5
7 | 5 | 5 | 4 | 0 | 14
8 | 9 | 9 | 10 | 0 | 28
9 | 16 | 16 | 19 | 1 | 52

For the purposes of this discussion, we will label the fully fixed functions
as $\mathfrak{f}_{\mathfrak{c}}$ and the partially fixed functions as
$\hat{\mathfrak{f}}_{\mathfrak{c}}$, where partially fixed means that they
have been constrained on $\mathfrak{c}$-type color-kinematics duality, but
that they have not yet been fixed such that their $\tilde{c}$-striated ordered
amplitudes $A_{\mathfrak{c}}^{\text{HD}}$ obey the adjoint-dependent $2$-basis
BCJ amplitude relations.

####  7.2.1 Doubly-dual amplitudes with permutation-invariant color

For permutation-invariant color $\mathfrak{c}=d^{5}$, finding a permutation
invariant $\hat{\mathfrak{f}}_{\mathfrak{p}}\\!\left(s_{ij},\tilde{c}\right)$
function is simple from our building blocks:

$\hat{\mathfrak{f}}_{\mathfrak{p}}^{(n)}=\left({\mathfrak{a}^{(n)}}\,\small$\mathfrak{p}$⃝\,{\tilde{c}^{\mathfrak{a}}}\right)=\sum_{g\in\Gamma_{3}}^{15}\mathfrak{a}^{(n)}_{g}\tilde{c}^{\mathfrak{a}}_{g}$
(167)

This is simply the composition of adjoint scalar kinematic numerators with
adjoint color to form a permutation invariant. We emphasize that all of thee
adjoint scalar kinematic numerators are constructible via iterated composition
of $\mathfrak{r}^{(1)}$. Only this single unit-step scalar is fundamentally
required.

In general, there can be multiple such permutation invariants at any given
mass dimension; these are taken to be summed over, with each given an
independent coefficient, thus defining the relevant candidate solution
$\hat{\mathfrak{f}}_{\mathfrak{p}}^{(n)}$. One then constructs the candidate
bi-color amplitude as:

$\mathcal{A}^{\text{bi-
color}}=d^{5}\,\hat{\mathfrak{f}}_{\mathfrak{p}}^{(n)}\\!\left(s_{ij},\tilde{c}\right)\equiv\sum_{\sigma\in
S_{4}}\textrm{Tr}(\widetilde{T}^{a_{1}}\widetilde{T}^{a_{\sigma_{1}}}\cdots\widetilde{T}^{a_{\sigma_{4}}})A_{\mathfrak{p}}^{\text{HD}}(1\sigma)$
(168)

These free coefficients within $\hat{\mathfrak{f}}_{\mathfrak{p}}$ are fixed
by requiring that the $\tilde{c}$-striated ordered amplitudes
$A_{\mathfrak{p}}^{\text{HD}}=A_{\mathfrak{p}}^{\text{HD}}\left(s_{ij},d^{5}\right)$
satisfy the adjoint amplitude relations. Any remaining coefficients represent
distinct valid local operator predictions at this mass-dimension.

We note that $\mathfrak{f}_{\mathfrak{p}}^{(n)}$ will generate the
$\mathcal{O}\left(\left({\alpha^{\prime}}\right)^{n+2}\right)$ order
correction to the theory. Orders in the dimensionful parameter
${\alpha^{\prime}}$ track the order in higher-derivative correction via the
number of additional scalar Lorentz invariants in the amplitude relative to
the uncorrected theory, as discussed in the context of eq. (56). Our
manifestly local construction of $\mathfrak{f}_{\mathfrak{c}}$ containing $n$
dot-products implies that the corresponding adjoint numerators $c^{\text{HD}}$
must contain $(n+2)$ dot-products in order to compensate for the two
propagators, versus zero such dot-products in the uncorrected bi-adjoint
theory numerators.

For $n<7$, the $2$-basis adjoint relation of eq. (166) admits only vanishing
solutions for $\mathfrak{f}_{\mathfrak{p}}^{(n)}$; for $n=7$, or ninth order
in ${\alpha^{\prime}}$, there is one solution, and at tenth order in
${\alpha^{\prime}}$, there are two doubly dual local amplitudes with $d^{5}$
permutation invariant color. This high mass dimension makes it particularly
unwieldy to give even a simple example. We will sketch an example of the
procedure for identifying this lowest order non-vanishing
$\mathfrak{f}_{\mathfrak{p}}^{7}$, called $\tt{fPI[9]}$ in the associated
ancillary files ancFiles .

As previously discussed in Table 2, there are 14 adjoint scalar solutions
$\mathfrak{a}^{7}_{i}$ at seven orders in Lorentz-invariant dot products.
These have been saved as a sum over distinct coefficients in ${\tt
adjoint[7]}$ in the ancillary files ancFiles . We may write the candidate
$\hat{\mathfrak{f}}_{\mathfrak{p}}^{(7)}$ as follows:

$\hat{\mathfrak{f}}_{\mathfrak{p}}^{(7)}=\sum_{i=1}^{14}b_{i}\left({\tilde{c}}\,\small$\mathfrak{p}$⃝\,{\mathfrak{a}^{7}_{i}}\right)\,,$
(169)

where each $b_{i}$ is an ansatz coefficient we give to the
composition111Recall as per eq. (167) that this composition corresponds to
summing over the 15 cubic graphs, dressing each with a product of both adjoint
weights. of the $i^{\text{th}}$ adjoint building block with $\tilde{c}$ to
form a permutation invariant. The candidate expression for the full amplitude
is then given as the product of this ansatz with $d^{abcde}$.

We may constrain this to be a doubly-dual local solution by demanding that the
coefficients of $\tilde{c}$ satisfy the $2$-basis BCJ amplitude relations.
Specifically, after expressing $\tilde{c}$ in a spanning DDM basis where legs
one and five are fixed, $\tilde{c}(1\sigma 5)$, the coefficient of each
$\tilde{c}$ constitutes an ordered amplitude. We constrain this candidate
expression for the ordered amplitude $A_{\mathfrak{p}}^{\text{HD}}(1\sigma 5)$
such that eq. (166) holds, fixing all but one coefficient as follows:

$\left\\{b_{1}\to
0,b_{2}\to-\frac{25b_{13}}{4},b_{3}\to\frac{15b_{13}}{4},b_{4}\to\frac{3b_{13}}{4},b_{5}\to-\frac{b_{13}}{4},b_{6}\to-\frac{3b_{13}}{4},\right.\\\
\left.b_{7}\to 0,b_{8}\to 0,b_{9}\to
0,b_{10}\to-\frac{b_{13}}{4},b_{11}\to-\frac{3b_{13}}{2},b_{12}\to-\frac{3b_{13}}{4},b_{14}\to
0\right\\}\,.$ (170)

The lone remaining parameter, $b_{13}$, can be identified as the Wilson
coefficient of the corresponding local five-point operator. In the case where
there are multiple unrelated solutions, such as order ten in
${\alpha^{\prime}}$, we have multiple satisfactory operators that can each
rightfully have its own Wilson coefficient.

####  7.2.2 Doubly-dual amplitudes with hybrid color

For hybrid color $\mathfrak{c}=c_{\mathfrak{h},1}=d^{4}f^{3}$, we similarly
construct hybrid $\mathfrak{f}_{\mathfrak{h}}$ functions from scalar
kinematics and the secondary $\tilde{c}$ adjoint color factors. It turns out
to be sufficient to consider nested compositions just like those used for
factorizing amplitudes:

$\hat{\mathfrak{f}}_{\mathfrak{h}}^{(n)}\equiv\left(\mathfrak{r}^{(1)}\,$\mathfrak{h}$⃝\,\mathfrak{f}_{\mathfrak{h}}^{(n-1)}\right)$
(171)

where for $n=1$, we define
$\hat{\mathfrak{f}}_{\mathfrak{h}}^{(1)}\equiv\left(\mathfrak{r}^{(1)}\,$\mathfrak{h}$⃝\,\tilde{c}_{\mathfrak{a}}\right)$.
Again, generically we find multiple distinct solutions to the hybrid
conditions at each mass dimension, which will be labeled with independent
coefficients and summed over. The corresponding ordered amplitudes
$A_{\mathfrak{h}}^{\text{HD}}=A_{\mathfrak{h}}^{\text{HD}}\left(s_{ij},d^{4}f^{3}\right)$
are then constrained on the 2-basis field-theory relations of eq. (166) to
find the fully fixed $\mathfrak{f}_{\mathfrak{h}}^{(n)}$ functions. The full
amplitude is then given by:

$\mathcal{A}=\sum_{g\in\Gamma{}_{\mathfrak{h}}}d^{4}f^{3}(g)\mathfrak{f}_{\mathfrak{h}}(g)=\sum_{\sigma\in
S_{4}}\textrm{Tr}(\widetilde{T}^{a_{1}}\widetilde{T}^{a_{\sigma_{1}}}\cdots\widetilde{T}^{a_{\sigma_{4}}})A_{\mathfrak{h}}^{\text{HD}}(1\sigma)$
(172)

where the first sum runs over the ten graphs $\Gamma{}_{\mathfrak{h}}$ with
one cubic vertex and one quartic vertex, and the second sum over the basis of
traces at five points. This first expression makes manifest hybrid color-
kinematics duality through the local hybrid-dual
$\mathfrak{f}_{\mathfrak{h}}^{(n)}$ graph weights; the second representation
exhibits adjoint color-kinematics duality through its eq. (166) satisfying
ordered amplitudes.

For the permutation-invariant doubly dual amplitudes, we find a need to fix
each individual order’s $\hat{\mathfrak{f}}_{\mathfrak{c}}$ function such that
the $\tilde{c}$-ordered amplitudes satisfy the $2$-basis relations of eq.
(166). Hybrid solutions turn out to be delightfully simpler—we can compose
directly from BCJ-compatible solution to BCJ compatible solution. To see this,
let us run through some examples at low mass dimension.

Our first objects of interest are the linear nested composition,
$\hat{\mathfrak{f}}_{\mathfrak{h}}^{(1)}\equiv\left(\mathfrak{r}^{(1)}\,$\mathfrak{h}$⃝\,\tilde{c}_{\mathfrak{a}}\right)$,
and the quadratic nested composition,
$\hat{\mathfrak{f}}_{\mathfrak{h}}^{(2)}=\left(\mathfrak{r}^{(1)}\,$\mathfrak{h}$⃝\,\hat{\mathfrak{f}}_{\mathfrak{h}}^{(1)}\right)$;
the corresponding ordered amplitudes in both cases admit no solutions that
satisfy the 5-point fundamental BCJ relation. The cubic nested composition
$\hat{\mathfrak{f}}_{\mathfrak{h}}^{(3)}\equiv\left(\mathfrak{r}^{(1)}\,$\mathfrak{h}$⃝\,\hat{\mathfrak{f}}_{\mathfrak{h}}^{(2)}\right)$
contains eight hybrid-color-dual building blocks: both composition rules used,
$\left(\mathfrak{r}^{(1)}\,$\mathfrak{h}$⃝\,\mathfrak{h}\right)$ or
$\left(\mathfrak{r}^{(1)}\,$\mathfrak{h}$⃝\,\mathfrak{a}\right)$, generate two
independent solutions, so there is, for instance, two building blocks within
$\hat{\mathfrak{f}}_{\mathfrak{h}}^{(1)}$. By third order, we have $2^{3}=8$
building blocks, which we label as $h^{(3)}_{i}$. We may write the candidate
$\hat{\mathfrak{f}}_{\mathfrak{h}}^{(3)}$ as follows:

$\hat{\mathfrak{f}}_{\mathfrak{h}}^{(3)}=\sum_{i=1}^{8}b_{i}h^{(3)}_{i}(\tilde{c},s_{jk})\,.$
(173)

Here, $b_{i}$ are coefficients of the eight distinct hybrid building blocks
$h^{(3)}_{i}$ that have been constructed via composition out momentum
invariants and $\tilde{c}$. These $b_{i}$ are to be fixed on the 2-basis BCJ
relations at five-points of eq. (166) to construct any doubly-dual solutions,
if possible. We build the full amplitude by summing over all ten relevant
graphs $\Gamma_{\mathfrak{h}}$, each dressed with $d^{4}f^{3}$ and our
constructed hybrid-dual $\hat{\mathfrak{f}}_{\mathfrak{h}}^{(3)}$. After
expressing the $\tilde{c}$ in a DDM basis $\tilde{c}(1\sigma 5)$, we may
interpret the coefficient of each color weight as an ordered amplitude. These
ordered amplitudes must be fixed to satisfy eq. (166) in order to represent
the fully doubly-dual solution,

$\mathcal{A}=\sum_{g\in\Gamma{}_{\mathfrak{h}}}d^{4}f^{3}(g)\mathfrak{f}_{\mathfrak{h}}(g)=\sum_{\sigma\in
S_{3}}\tilde{c}(1\sigma 5)A_{\mathfrak{h}}^{\text{HD}}(1\sigma 5)\,.$ (174)

Imposing eq. (166) on the candidate expressions for
$A_{\mathfrak{h}}^{\text{HD}}$ constrains all but one of the $b_{i}$
parameters, yielding a single solution at this order. The unconstrained
parameter should be understood as a stand-in for the Wilson coefficient
associated with the corresponding higher-derivative local operator. We label
this solution by the number of explicit Lorentz invariants appearing alongside
$\tilde{c}$ in the local hybrid weight, $\mathfrak{f}_{\mathfrak{h}}^{(3)}$.
This gives rise to a doubly dual contact amplitude at fifth order in
${\alpha^{\prime}}$, as the corresponding higher-derivative adjoint color
weight $c^{\text{HD}}$ for this amplitude expressed over a sum of cubic graphs
must contain five dot-products to compensate for the two cubic propagators. We
therefore call this solution ${\tt fHybrid[5]}$ in the ancillary machine
readable files ancFiles , where we index by the order in ${\alpha^{\prime}}$.

At the next order, one could calculate
$\hat{\mathfrak{f}}_{\mathfrak{h}}^{(4)}\equiv\left(\mathfrak{r}^{(1)}\,$\mathfrak{h}$⃝\,\hat{\mathfrak{f}}_{\mathfrak{h}}^{(3)}\right)$
and then fix the ordered amplitude to generate the final solution
$\mathfrak{f}_{\mathfrak{h}}^{(4)}$, but there is a much more direct route:
composing the fully fixed cubic solution $\mathfrak{f}_{\mathfrak{h}}^{(3)}$
with $\mathfrak{r}^{(1)}$:

$\mathfrak{f}_{\mathfrak{h}}^{(4)}\equiv\left(\mathfrak{r}^{(1)}\,$\mathfrak{h}$⃝\,\mathfrak{f}_{\mathfrak{h}}^{(3)}\right)$
(175)

This composition $\mathfrak{f}_{\mathfrak{h}}^{(4)}$ already yields ordered
amplitudes that automatically satisfy the $(m-3)!$ relations, seemingly by
virtue of using the fully fixed $\mathfrak{f}_{\mathfrak{h}}^{(3)}$ in the
composition!

At the next order of higher derivative correction, we may play the same sort
of game, only now we have two building blocks at our disposal,
$\mathfrak{f}_{\mathfrak{h}}^{(3)}$ and $\mathfrak{f}_{\mathfrak{h}}^{(4)}$:

$\mathfrak{f}_{\mathfrak{h}}^{(5)}\equiv\left\\{\left(\mathfrak{r}^{(1)}\,$\mathfrak{h}$⃝\,\mathfrak{f}_{\mathfrak{h}}^{(4)}\right),\,\mathfrak{f}_{\mathfrak{h}}^{(3)}\,\mathfrak{p}^{(2)}\right\\}$
(176)

giving rise to five independent doubly dual amplitudes at seventh order in
${\alpha^{\prime}}$. We find that this pattern continues. It is sufficient to
simply consider compositions and products with permutation invariants of
lower-order fully fixed $\mathfrak{f}_{\mathfrak{c}}$ functions to find
higher-order fully fixed $\mathfrak{f}_{\mathfrak{c}}$ functions through at
least ninth order in ${\alpha^{\prime}}$,

$\mathfrak{f}_{\mathfrak{h}}^{(n)}\equiv\left\\{\left(\mathfrak{r}^{(1)}\,$\mathfrak{h}$⃝\,\mathfrak{f}_{\mathfrak{h}}^{(n-1)}\right),\,\mathfrak{f}_{\mathfrak{h}}^{(n-2)}\,\mathfrak{p}^{(2)}\right\\}\,,$
(177)

without any need to fix the adjoint-striated ordered amplitudes at each order
on the $(m-3)!$ constraints, as they are already satisfied! Through
$\mathcal{O}\left(\left({\alpha^{\prime}}\right)^{9}\right)$, we have
confirmed via explicit ansatz calculations that our composition structures and
doubly dual local construction give rise to all possible contact amplitudes
containing scalar higher-derivative corrections and hybrid color, with just
the linear relaxed solution $\mathfrak{r}^{(1)}$ and the relevant color
structures as input building blocks. Although we do not currently have a
proof, we expect this simple pattern to continue.

####  7.2.3 Doubly-dual amplitudes with adjoint color

Once hybrid $\mathfrak{f}_{\mathfrak{h}}$ functions are known, obtaining
adjoint $\mathfrak{f}_{\mathfrak{a}}$ functions for
$\mathfrak{c}=c_{\mathfrak{a},1}=f^{3}f^{3}f^{3}$ is rendered trivial via
casting:

$\mathfrak{f}_{\mathfrak{a}}^{(n)}=\mathfrak{a}\left[\mathfrak{f}_{\mathfrak{h}}^{(n)}\right]\,.$
(178)

Explicitly, in terms of individual graph labels,

$\displaystyle\mathfrak{f}_{\mathfrak{a}}(12345)=-\frac{1}{5}\Big{(}2\,\mathfrak{f}_{\mathfrak{h}}(12345)+\mathfrak{f}_{\mathfrak{h}}(12435)-\mathfrak{f}_{\mathfrak{h}}(12534)$
(179)
$\displaystyle+2\,\mathfrak{f}_{\mathfrak{h}}(23415)-2\,\mathfrak{f}_{\mathfrak{h}}(23514)\Big{)}$

The process could also be repeated in the opposite order, finding adjoint
$\mathfrak{f}_{\mathfrak{c}}$ functions first via composition and then casting
to hybrid solutions. This one-to-one correspondence means there are equal
numbers of doubly dual contact amplitude solutions with hybrid color and with
adjoint color at each order in higher-derivative correction.

The combined methods of composition, casting, and doubly dual local
construction dramatically simplify calculation – at ninth order in
${\alpha^{\prime}}$, for example, an ansatz for $c^{\text{HD}}$ would contain
6435 free parameters. Introducing the notion of doubly dual amplitudes would
allow us to write a smaller ansatz down for $\mathfrak{f}_{\mathfrak{a}}$,
with 1980 free parameters. Using composition, we are able to avoid ansatze
entirely: the hybrid solution $\mathfrak{f}_{\mathfrak{h}}$ is obtained
directly by composing $\mathfrak{r}^{(1)}$ with the previous order in
correction, with no parameters needing to be fixed, and then the adjoint
solution $\mathfrak{f}_{\mathfrak{a}}$ is written down immediately by casting
$\mathfrak{f}_{\mathfrak{h}}$ to adjoint.

####  7.2.4 Doubly-dual amplitudes with sandwich color

One can proceed in much the same way for sandwich color
$\mathfrak{c}=c_{\mathfrak{s}}=f^{3}d^{3}f^{3}$. As its constraints are looser
(obeying a six-term identity rather than stricter three-term Jacobi
identities), a more complicated composition pattern is needed to span all
solutions. These compositions involve not only the $\mathfrak{r}^{(1)}$
building block, but also the quadratic sandwich building block
$\mathfrak{s}^{(2)}=\left(\mathfrak{r}^{(1)}\,$\mathfrak{s}$⃝\,\mathfrak{r}^{(1)}\right)$
built from the linear relaxed solution $\mathfrak{r}^{(1)}$.

###  7.3 Restricting to bi-adjoint and higher-derivative momentum kernels

All of the higher-derivative correction amplitudes discussed in this and the
previous section admit a double-copy description via standard KLT. In this
description, the standard field-theory KLT momentum kernel (which does not
itself carry any higher-derivative corrections) facilitates the marriage of
ordered amplitudes involving the adjoint higher-derivative color weights
$c^{\text{HD}}$ with ordered amplitudes involving secondary adjoint color
factors $\tilde{c}$ (simply the singly-ordered bi-adjoint scalar theory
ordered amplitudes). It is these secondary ordered amplitudes which can be
replaced with ordered vector amplitudes to generate higher-derivative
corrections to Yang-Mills, as we will discuss in the next section.

Before we do so, let us recall to the reader that there exists a notion of
local higher-derivative momentum kernels introduced by Chi, Elvang,
Herderschee, Jones, and Paranjape222See, e.g. slides and video at ref.
callumTalk .. Here we identify an additional strategy one might employ towards
identifying these types of local adjoint-compatible momentum kernels. We begin
by using our notion of local $\mathfrak{c}$-dual double-copy structure to
write down all the ways that adjoint color weights can be combined with local
higher-derivative corrections:

$\mathcal{A^{\text{bi-
adj+HD}}}=\sum_{\mathfrak{c}}\sum_{g\in\Gamma_{\mathfrak{c}}}\tilde{\mathfrak{f}}_{\mathfrak{c}}(g)\\!\left(s_{ij},\tilde{c}\right)~{}\mathfrak{f}_{\mathfrak{c}}(g)\\!\left(s_{ij},c\right)\,.$
(180)

Note that we sum over all algebraic structures $\mathfrak{c}$ relevant at five
points. For each structure $\mathfrak{c}$, we must dress the relevant graphs
$\Gamma_{\mathfrak{c}}$ with the product between two $\mathfrak{c}$-dual
functions, $\tilde{\mathfrak{f}}_{\mathfrak{c}}(g)$ and
$\mathfrak{f}_{\mathfrak{c}}(g)$. Each function is dependent upon adjoint-type
color, $c$ or $\tilde{c}$, as well as scalar kinematics that conspire with the
adjoint color to allow the entire weight $\mathfrak{f}_{\mathfrak{c}}$ to be
compatible with $\mathfrak{c}$-type algebraic constraints – in a manner
compatible with adjoint-type BCJ relations, just as these
$\mathfrak{f}_{\mathfrak{c}}$ functions do in our doubly dual local
amplitudes. This constitutes essentially a double copy of a doubly-dual local
amplitude with itself, with the appropriate color-dual replacement
$\mathfrak{c}(g)\rightarrow\tilde{\mathfrak{f}}_{\mathfrak{c}}(g)\\!\left(s_{ij},\tilde{c}\right)$,
giving the most general local higher-derivative modification of the bi-adjoint
scalar to admit a local double copy description.

We then rewrite this amplitude, at each order in mass dimension, by expressing
every adjoint color factor $c$ in a basis of half-ladder color weights,
$c(1|\sigma|5)$, and likewise for $\tilde{c}$. This allows us to define
doubly-ordered amplitudes of this theory, c.f. related identification of
doubly-ordered ref. Carrasco:2019qwr , by identifying them as the unique
coefficients of these independent color-basis monomials:

$\mathcal{A^{\text{bi-adj+HD}}}=\sum_{\sigma,\rho\in
S_{n-2}}\tilde{c}(1|\sigma|n)A^{\text{bi-
adj+HD}}(\sigma|\rho)c(1|\rho|n)=\tilde{c}^{T}\cdot A^{\text{bi-adj+HD}}\cdot
c\,.$ (181)

We now exploit the standard KLT kernel, $S$, in order to rewrite $c$ and
$\tilde{c}$ in terms of their $(n-3)!$ basis of (singly-)ordered amplitudes,
$c=S\cdot A$ and $\tilde{c}=S\cdot\tilde{A}$,

$\mathcal{A^{\text{bi-adj+HD}}}=\tilde{A}^{T}\cdot S^{T}\cdot A^{\text{bi-
adj+HD}}\cdot S\cdot A\,.$ (182)

Finally, we can now identify the most general higher-derivative kernel
compatible with a dual-adjoint striation at this mass-dimension:

$S^{\text{HD}}=S^{T}\cdot A^{\text{bi-adj+HD}}\cdot S\,,$ (183)

so that this most general local higher-derivative correction amplitude to the
bi-adjoint scalar may equivalently be expressed in terms of uncorrected
ordered amplitudes $A$ and this higher-derivative momentum kernel
$S^{\text{HD}}$,

$\mathcal{A^{\text{bi-adj+HD}}}=\tilde{A}^{T}\cdot S^{\text{HD}}\cdot A\,.$
(184)

We identify some intriguing open questions regarding such an approach in our
conclusion section.

##  8 Double Copy to Gauge and Gravity Corrections

###  8.1 Higher-derivative gauge corrections

We have demonstrated how composition and casting can yield an infinite tower
of higher-derivative corrections in the form of adjoint-type modified color
weights dressing graphs in a bi-color scalar amplitude. The fact that Yang-
Mills amplitudes are compatible with maximal supersymmetry at tree-level means
that, through double-copy construction, we can trivially map bi-color higher-
derivative amplitudes to an infinite tower of higher-derivative corrections to
Yang-Mills – all of them entirely compatible with maximal supersymmetry. The
bi-colored amplitudes we constructed in the previous section have the
following form,

$\mathcal{A}_{5}^{\text{BC+HD}}=\sum_{g}\frac{\tilde{c}_{g}c^{\text{HD}}_{g}}{d_{g}}\,.$
(185)

By virtue of the fact that the higher-derivative color weights,
$c^{\text{HD}}$, satisfy the adjoint algebraic relations in eqs. (74) and
(77), we are free to replace the secondary adjoint color $\tilde{c}$ with any
graph weight that depends on adjoint relations to build a gauge-invariant
amplitude, like the kinematic numerators of Yang-Mills theory, or indeed any
vector weight that can be expressed in terms of adjoint color-dual numerators!
For example, higher-derivative corrections compatible with maximally
supersymmetric Yang-Mills can be generated by pairing $c^{\text{HD}}$ with the
vector weight $n_{g}^{\text{sYM}}$ associated with Yang-Mills at five-points,

$\mathcal{A}_{5}^{\text{vec}+\text{HD}}=\sum_{g}\frac{n_{g}^{\text{vec}}c^{\text{HD}}_{g}}{d_{g}}\,,$
(186)

with $n_{g}^{\text{vec}}=n_{g}^{\text{sYM}}$. The resulting amplitudes will be
compatible with maximal supersymmetry and manifest an adjoint double copy
structure, with the additional mass-dimension and generic color-weights from
the higher-derivative corrections encoded in $c^{\text{HD}}$. For an example
calculation matching one of our higher-derivative vector amplitudes (generated
via double copy of Yang-Mills with our second-order modified color weights
$c^{\text{HD}}$) to a traditional Feynman rules calculation starting from
higher-derivative operators in the action, see Appendix C.

Let us point out a feature at this stage. Because we have striated along
adjoint structures, it is entirely sufficient to build the two independent
ordered amplitudes, $A_{5}^{\text{BC+HD}}(12345)$ and
$A_{5}^{\text{BC+HD}}(13245)$, and simply use the standard KLT relation with
ordered vector amplitudes $A_{5}^{\text{vec}}(12354)$ and
$A_{5}^{\text{vec}}(13254)$ to arrive at higher-derivative gauge theory
amplitudes (equivalent to those obtained via graph-based adjoint double copy).
Let us emphasize application of the adjoint momentum kernel is possible even
with non-adjoint color weights like $d^{abcde}$ and $f^{3}d^{3}f^{3}$ because
we have composed those colors with scalar weights to form adjoint-type
amplitudes: by virtue of the composition rules, non-adjoint color conspires
with scalar kinematics to satisfy adjoint constraints. This adjoint-structure
for the $\tilde{c}$ weights guarantees that the replacement with adjoint-dual
vector weights $\tilde{c}\rightarrow n^{\text{vec}}_{\mathfrak{a}}$ (such as
$n^{\text{sYM}}$ results in gauge-invariant amplitudes even in the dual-
striated local representation,

$\mathcal{A}^{\text{vec+HD}}=\sum_{g\in\Gamma_{\mathfrak{c}}}\mathfrak{c}_{g}~{}\mathfrak{f}_{\mathfrak{c}}(g)\\!\left(s_{ij},n^{\text{vec}}_{\mathfrak{a}}\right)\,.$
(187)

It is beyond the scope of this current paper to identify all distinct adjoint
vector weights, modulo scalar-weight modifications, at five-points. While
there were only eight at four-points, there is more freedom at five-points. It
will be interesting and useful to identify the minimal set of vector building
blocks required to build all adjoint-type higher-derivative vector
corrections. We provide some examples, including Yang-Mills, in ancillary
Mathematica files ancFiles .

###  8.2 Higher-derivative gravity corrections

The care we must take in promoting the above discussion to a double-copy
construction to gravity is simply to ensure that gauge-invariance is manifest
for both copies after replacing the remaining color-weights in $c^{\text{HD}}$
of eq. (186), or equivalent the $\mathfrak{c}_{g}$ in eq. (187), with vector
weights. If we can maintain gauge-invariance of both copies of vector weights,
our resulting expression will describe higher-derivative corrections to the
scattering of gravitons.

For trivially modified color weights $c^{\text{HD}}(g)\propto
f^{3}f^{3}f^{3}(g)\times\mathfrak{p}$, we can simply replace
$f^{3}f^{3}f^{3}(g)\to\tilde{n}^{\text{vec}}(g)$ in eq. (186) as long as the
$n^{\text{vec}}_{g}$ have been expressed in a manifestly adjoint-color-dual
form. This is because permutation invariants will not endanger the
$n^{\text{vec}}$’s ability to ensure the gauge-invariance of the
$\tilde{n}^{\text{vec}}$ copy. As $c^{\text{HD}}$ is adjoint-type the gauge
invariance of the $n^{\text{vec}}$ copy is already manifest.

For more general modified color-weights, we must replace the color weights
$\mathfrak{c}$ within $c^{\text{HD}}(\mathfrak{c},s_{ij})$ with vector weights
both obey the same algebra as $\mathfrak{c}$ and require the algebra of
$\mathfrak{c}$ to build gauge-invariant quantities. In factorizing
corrections, the color $\mathfrak{c}$ within the modified color weights
$c^{\text{HD}}$ is either of type $f^{3}f^{3}f^{3}$ or $d^{4}f^{3}$, so we may
only replace these, respectively, with adjoint ($\mathfrak{a}_{{\text{vec}}}$)
or hybrid ($\mathfrak{h}_{{\text{vec}}}$) vector weights that respect gauge
invariance of the overall amplitude. For sufficiently high mass-dimension, it
is possible for the individual vector graph weights to be gauge invariant,
rendering the question of gauge invariance of the full amplitude trivial.
Examples of such local dual vector weights are
$\mathfrak{a}^{5}_{{\text{vec}}}$, $\mathfrak{h}^{5}_{{\text{vec}}}$,
$\mathfrak{s}^{6}_{{\text{vec}}}$, and $\mathfrak{p}^{8}_{{\text{vec}}}$,
where the superscript here refers to the number of Lorentz-invariant dot
products per term in the numerator (for comparison, Yang-Mills numerators
contain four dots per term, but they do not fall into this category, as the
graph weights are not themselves gauge-invariant; rather, they conspire
between graph channels to cancel out any gauge dependence in the overall
amplitude). Useful in both factorizing and local gauge corrections, and the
predictions associated with local gravity counterterms to be discussed, we
provide these example local gauge-invariant algebraic vector weights in
ancillary Mathematica files ancFiles .

This strategy is particularly clear in the dual-striated form of the higher-
derivative vector amplitude, eq. (187). If we are to promote
$\mathfrak{c}_{g}$ in eq. (187) to encode spin-1 vector information via
double-copy replacement, the vector weights
$\tilde{n}^{\text{vec}}_{\mathfrak{c}}$ must satisfy the same algebraic
relations as $\mathfrak{c}$ and must result in gauge-invariant amplitude
expressions when dressing the graphs relevant to that algebra, such that
$\mathcal{A}=\sum_{g\in\Gamma_{\mathfrak{c}}}\mathfrak{c}(g)\tilde{n}^{\text{vec}}_{\mathfrak{c}}(g)$
is gauge-invariant. If such a vector weight can be identified, its
corresponding diffeomorphism-invariant gravity amplitude may be written down
as:

$\mathcal{A}^{\text{grav}}=\sum_{g\in\Gamma_{\mathfrak{c}}}\tilde{n}^{\text{vec}}_{\mathfrak{c}}(g)~{}\tilde{n}^{\text{vec}}_{\mathfrak{c}}(g)\,,$
(188)

and its associated higher-derivative gravity corrections are found simply
through double copy with the higher-derivative gauge theory corrections
$\mathcal{A}^{\text{vec+HD}}$ given by eq. (187):

$\mathcal{A}^{\text{grav+HD}}=\sum_{g\in\Gamma_{\mathfrak{c}}}\tilde{n}^{\text{vec}}_{\mathfrak{c}}(g)~{}\mathfrak{f}_{\mathfrak{c}}(g)\\!\left(s_{ij},n^{\text{vec}}_{\mathfrak{a}}\right)\,.$
(189)

Finally, we remark on one final form of local higher-derivative gravity
counterterms that may be obtained from our doubly-dual construction: for color
structure $\mathfrak{c}$, the dual properties of $\mathfrak{f}_{\mathfrak{c}}$
ensure that we may make the replacement
$\mathfrak{c}(g)\rightarrow\tilde{\mathfrak{f}}_{\mathfrak{c}}(g)\\!\left(s_{ij},\tilde{c}\right)$
to arrive at higher-derivative corrections to the bi-adjoint scalar. The most
general such form of these corrections, as discussed in the context of eq.
(180), is given by summing over all possible structures $\mathfrak{c}$:

$\mathcal{A^{\text{bi-
adj+HD}}}=\sum_{\mathfrak{c}}\sum_{g\in\Gamma_{\mathfrak{c}}}\tilde{\mathfrak{f}}_{\mathfrak{c}}(g)\\!\left(s_{ij},\tilde{c}\right)~{}\mathfrak{f}_{\mathfrak{c}}(g)\\!\left(s_{ij},c\right)\,.$
(190)

For each algebraic structure $\mathfrak{c}$, we sum over the relevant graphs g
in $\Gamma_{\mathfrak{c}}$, dressing each with two higher-derivative
$\mathfrak{c}$-dual graph weights, $\mathfrak{f}_{\mathfrak{c}}(g)$ and
$\tilde{\mathfrak{f}}_{\mathfrak{c}}(g)$. Since such
$\mathfrak{f}_{\mathfrak{c}}$ functions are fixed so that striating the
amplitude along the basis elements of the adjoint color weights $c$ yields
$(m-3)!$-compatible ordered amplitudes, the replacement $c\rightarrow
n^{\text{vec}}_{\mathfrak{a}}$ automatically yields gauge-invariant amplitudes
(as seen previously in eq. (187)). This compatibility with the $(m-3)!$
relations for ordered amplitudes of theories with adjoint-double-copy
structure is ensured by the algebraic properties of $\mathfrak{c}$, meaning
that this compatibility is preserved when $\mathfrak{c}$ is replaced with an
appropriately color-dual $\mathfrak{f}_{\mathfrak{c}}$ function. As a result,
both adjoint color weights within $\mathcal{A^{\text{bi-adj+HD}}}$, $c$ and
$\tilde{c}$, may be replaced with any adjoint-color-dual vector weights to
yield fully diffeomorphism-invariant higher-derivative local contact gravity
amplitudes:

$\mathcal{A^{\text{GR+HD}}}=\sum_{\mathfrak{c}}\sum_{g\in\Gamma_{\mathfrak{c}}}\tilde{\mathfrak{f}}_{\mathfrak{c}}(g)\\!\left(s_{ij},\tilde{n}^{\text{vec}}_{\mathfrak{a}}\right)~{}\mathfrak{f}_{\mathfrak{c}}(g)\\!\left(s_{ij},n^{\text{vec}}_{\mathfrak{a}}\right)\,,$
(191)

where these vector weights need not arise from the same theory; the only
condition is that both be adjoint-color-dual. We are free to choose both
vector weights to be from super-Yang-Mills, so that:

$\mathcal{A^{\text{SUGRA+HD}}}=\sum_{\mathfrak{c}}\sum_{g\in\Gamma_{\mathfrak{c}}}\tilde{\mathfrak{f}}_{\mathfrak{c}}(g)\\!\left(s_{ij},n^{\text{sYM}}\right)~{}\mathfrak{f}_{\mathfrak{c}}(g)\\!\left(s_{ij},n^{\text{sYM}}\right)\,,$
(192)

generates local contact higher-derivative corrections to supergravity. We
continue to label the dual graph weights $\tilde{\mathfrak{f}}_{\mathfrak{c}}$
and $\mathfrak{f}_{\mathfrak{c}}$ distinctly in this case to remind the reader
that, even though functions obey the same algebraic relations and are
functions of the same vector weight $n^{\text{sYM}}$, they need not be the
same, as they may correspond to higher-derivative corrections from different
orders in mass-dimension.

##  9 Ladder to String Theory

We expect that our new five-point color-kinematic solutions (constructively
built from color-weights and ultimately the unit-step relaxed scalar solution)
span the whole space of BCJ-compatible corrections to Yang-Mills and, in
particular, contain the open superstring corrections. The connection can be
made through the bi-colored Z-theory amplitudes, which lift, through standard
field-theory double copy, field-theoretic amplitudes to string theory
amplitudes at tree-level Broedel2013tta ; Huang:2016tag ; Carrasco2016ldy ;
Mafra2016mcc ; Johansson2017srf .

$A^{\textrm{open}}(\tau)=\sum_{\rho,\sigma\in
S_{n-3}}Z(\tau|1,\sigma,n,n-1)S[\sigma|\rho]A^{\mathrm{vec}}(1,\rho,n-1,n)\,,$
(193)

where $S$ is the field-theory KLT momentum kernel and thus does not carry any
string higher-derivative corrections. $Z$-theory encodes all the
$\alpha^{\prime}$ corrections to Yang-Mills that show up in the tree-level
open superstring, and carries color structure from two different gauge groups:
the Chan-Paton factors, which can in principle sum over all single-trace
contributions, and secondary adjoint color-weights which are e.g. replaced by
$n^{\text{sYM}}$ when double-copying to the open superstring amplitude. These
color structures line up with $\mathfrak{c}$ and $\tilde{c}$, respectively, in
our constructed bicolored scalar amplitudes. In terms of the modified color
factors, the full $Z$-theory amplitudes can also be expressed as:

$\displaystyle\mathcal{Z}=\sum_{g}\frac{\tilde{c}_{g}c_{g}^{\textrm{HD}}}{d_{g}}\,,$
(194)

so the modified color factors $c^{\text{HD}}$ carry the ${\alpha^{\prime}}$
string corrections and the Chan-Paton color factors. Doubly-ordered amplitudes
$Z(\alpha|\beta)$ can be obtained from eq. (194) by expanding both the color
structures into traces:

$\displaystyle\mathcal{Z}=\sum_{\sigma,\rho\in
S_{n-1}}\textrm{Tr}(\widetilde{T}^{a_{1}}\widetilde{T}^{a_{\sigma_{1}}}\cdots\widetilde{T}^{a_{\sigma_{n-1}}})\textrm{Tr}(T^{b_{1}}T^{b_{\sigma_{1}}}\cdots
T^{b_{\sigma_{n-1}}})Z(1,\sigma|1,\rho)\,.$ (195)

We have verified that up to and including order $(\alpha^{\prime})^{9}$, our
solutions span the low-energy expansion of the open superstring at five-
points. This may be checked directly by comparing our results to $Z$-theory
amplitudes built using Berends-Giele recursion Mafra2016mcc , or by fixing to
the following expression for string amplitudes, which is known to high order
Mafra2011nv ; Schlotterer2012ny ; Mafra2011nw :

$A^{\text{open}}(1,\rho,n-1,n)=\sum_{\sigma\in
S_{n-3}}{F_{\rho}}^{\sigma}A^{\text{vec}}(1,\sigma,n-1,n)\,.$ (196)

Comparing eqns. (196) and (193) we obtain Mafra2016mcc :

${F_{\rho}}^{\sigma}=\sum_{\tau\in
S_{n-3}}S[\tau|\sigma]Z(1,\rho,n-1,n|1,\tau,n,n-1)\,.$ (197)

For multiplicity $n=5$ and the choice $\rho=\\{2,3\\}$, this yields equations:

$\displaystyle{F_{23}}^{23}$
$\displaystyle=S[23|23]Z(12345|12354)+S[32|23]Z(12345|13254)$ (198)
$\displaystyle{F_{23}}^{32}$
$\displaystyle=S[23|32]Z(12345|12354)+S[32|32]Z(12345|13254)$ (199)

These equations will allow us to constrain our results to match string theory.
Our amplitudes, ordered on both group’s color factors, will be used in place
of the $Z$-theory doubly-ordered amplitudes in these equations, and results
for $F$ will be taken directly from string theory. $F$ can be written in terms
of $M$ and $P$ matrices Schlotterer2012ny ; Broedel:2013tta :

$\displaystyle F=$
$\displaystyle\;1+\zeta_{2}\;P_{2}+\zeta_{3}\;M_{3}+\zeta_{2}^{2}\;P_{4}+\zeta_{2}\zeta_{3}\;P_{2}M_{3}+\zeta_{5}\;M_{5}$
$\displaystyle+\zeta_{2}^{3}\;P_{6}+\frac{1}{2}\;\zeta_{3}^{2}\;M_{3}M_{3}+\;\zeta_{7}\;M_{7}+\zeta_{2}\zeta_{5}\;P_{2}M_{5}+\zeta_{2}^{2}\zeta_{3}\;P_{4}M_{3}$
$\displaystyle+\zeta_{2}^{4}\;P_{8}+\zeta_{3}\zeta_{5}\;M_{5}M_{3}+\frac{1}{2}\;\zeta_{2}\zeta_{3}^{2}\;P_{2}M_{3}M_{3}+\cdots$
(200)

We parameterize our solutions as follows: $b[n,i]$ labels the $i^{th}$
factorizing solution at $n^{th}$ order in ${\alpha^{\prime}}$, and
$b[n,\mathfrak{c},i]$ labels the $i^{th}$ local solution at $n^{th}$ order in
${\alpha^{\prime}}$, with color structure $\mathfrak{c}$ of type
$\mathfrak{a},\mathfrak{h},\mathfrak{s},$ or $\mathfrak{p}$. Here we reproduce
the results through fifth order of fixing our solutions to the
${\alpha^{\prime}}$ expansion of $Z$-theory (and thus equivalently fixing to
open superstring theory):

$\displaystyle\left(\alpha^{\prime}\right)^{2}:$ $\displaystyle\
b[2,1]=24\,\zeta_{2}\,,$ (201)
$\displaystyle\left(\alpha^{\prime}\right)^{3}:$ $\displaystyle\
b[3,1]=-8\,\zeta_{3}\,,$ (202)
$\displaystyle\left(\alpha^{\prime}\right)^{4}:$ $\displaystyle\
b[4,1]=48\,\zeta_{2}^{2},\quad b[4,2]=-16\,\zeta_{2}^{2}/5\,$ (203)
$\displaystyle\left(\alpha^{\prime}\right)^{5}:$ $\displaystyle\
b[5,1]=384\,\zeta_{2}\zeta_{3},\quad b[5,2]=-32\,\zeta_{5},$ (204)
$\displaystyle\ b[5,\mathfrak{h},1]=-6650112\,\zeta_{2}\zeta_{3}/2615,\quad
b[5,\mathfrak{a},1]=-30960\,\zeta_{5}/523\,,$ (205)

with results up to order $(\alpha^{\prime})^{9}$ available in the ancillary
files. Note that neither $d^{5}$ nor $f^{3}d^{3}f^{3}$-type color solutions
($\mathfrak{p}$ or $\mathfrak{s}$) contribute to the string amplitude, as
their single-trace Chan-Paton coefficient amplitudes would violate the odd-
point reflection identity (just as these amplitudes are incompatible with the
Yang-Mills reflection identity, as discussed in the context of eq. (129)). All
other building blocks are given a fixed Wilson coefficient by comparison to
string theory. Above and including sixth order in ${\alpha^{\prime}}$, we
frequently find that multiple linearly-independent solutions (of both the
factorizing and local contact varieties) are available, as summarized in
Tables 5 and 6. Since string theory only picks out one particular combination
of these multiple distinct solutions, there are a number of distinct
(linearly-independent) operators that string theory avoids at tree level – and
whose invariant predictions are spanned by our construction at five-points.

Some of these distinct directions in solution space may be related to higher
genus. It is possible to match to the $M^{\prime}$ matrices of reference
Green2013bza , local solutions which were found to be compatible with
monodromy and preserve the BCJ relations of amplitudes, but that do not show
up in string amplitudes at tree level. The lowest mass dimension example
occurs at $(\alpha^{\prime})^{7}$, where we find:

$\displaystyle M_{7}^{\prime}:\ b[7,\mathfrak{a},1]=-\frac{1}{256}\,.$ (206)

This and higher mass dimensions weights, through ninth order, are collected in
ancillary files ancFiles .

##  10 Conclusion

We have demonstrated that the wealth of higher-derivative algebraic structure
at five points compatible with adjoint-type double-copy can arise from a
simple linear scalar building block much like at four-points. Beyond this we
have a conjectured form for such linear blocks at all multiplicity, eq. (1).
We expect this to result in the constructive ability to write down functional
expressions for a rich variety of gauge and gravity higher-derivative
predictions at arbitrary multiplicity. We demonstrate in detail how this can
proceed at five-points, allowing one to methodically construct higher mass-
dimension numerators from lower order solutions and find a ladder of
supersymmetry-compatible higher-derivative corrections to gauge theory and
gravity. We exemplify the construction of such composition rules with four-
point examples in Appendix A, and give explicit results for all of our five-
point composition rules in Appendix D. We emphasize that the value of having
such composition rules is that they allows us to build infinite towers of
solutions to functional-constrains without needing to invoke an ansatz.

Adjoint-compatible higher-derivative gauge theory amplitudes at five points
arise via double-copy from bi-colored amplitudes constructed with explicit
higher-derivative adjoint-compatible color-weights:
$(\text{sYM+HD})=\text{sYM}\otimes(\text{BC+HD})$. This has the virtue of
allowing us to impose either factorization limits or locality conditions in a
simpler bicolored scalar theory. We find factorizing solutions by composing
adjoint color factors (containing both $f^{3}f^{3}f^{3}$ and $d^{4}f^{3}$
structures) repeatedly with $\mathfrak{r}^{(1)}$ to give candidate expressions
for $c^{\text{HD}}$, then fixing on factorization. We conjecture that, above
seventh order, no new nested compositions are needed to generate the candidate
expressions that may be fixed on factorization, as eighth and ninth order were
spanned by lower order structures times permutation invariants.

Local solutions can be found by fixing the same candidate expressions instead
on the condition that all possible residues of the amplitude vanish. We find
that the local bicolored scalar amplitudes are more efficiently constructed by
making manifest a novel form of color-kinematics duality, in which each
amplitude is manifestly local and displays two different algebraic striations
between color and kinematics. These amplitudes like the factorizing solutions,
contain two color structures: color factors $\mathfrak{c}$ that will survive
in the gauge theory amplitude, of the form $f^{3}f^{3}f^{3}$, $d^{4}f^{3}$,
$f^{3}d^{3}f^{3}$, or $d^{5}$, as well as a secondary standard adjoint color
factor $\tilde{c}_{\mathfrak{a}}$, which is replaced with $n^{\text{vec}}$ in
the double copy to gauge theory corrections. Coefficients of a basis element
of $\tilde{c}_{\mathfrak{a}}$ are ordered amplitudes that consist of scalar
kinematics and $\mathfrak{c}$ color and obey the $(m-3)!$-basis BCJ relations,
ensuring compatibility with adjoint color-kinematics duality. Coefficients of
the color factors $\mathfrak{c}$,
$\mathfrak{f}_{\mathfrak{c}}(\tilde{c}_{\mathfrak{a}},s_{ij})$, contain the
secondary gauge group, and are manifestly local objects that satisfy the same
algebraic constraints as the particular color structure of $\mathfrak{c}$. For
instance, if $\mathfrak{c}_{g}=d^{4}f^{3}(g)$, its coefficient functions
$\mathfrak{f}_{\mathfrak{c}}{}_{,g}$ obeys the same mixed vertex
(anti)symmetry and four point identities as $d^{4}f^{3}$. The amplitude is
thus doubly dual: it exhibits both adjoint and $d^{4}f^{3}$ color-kinematics
duality, depending upon striation.

Our bicolored scalar solutions can be trivially recycled to maximally
supersymmetric gauge theory corrections through double copy with super Yang-
Mills compatible graph-weights. Double copy of gauge theory corrections with
another gauge theory to generate higher derivative supersymmetry-compatible
gravity amplitudes is straightforward in some cases and more complicated in
others. In all amplitudes, the replacement of the second gauge group’s adjoint
color factor $\tilde{c}_{\mathfrak{a}}\rightarrow
n^{\text{vec}}_{\mathfrak{a}}$ is always allowed; but care must be taken when
replacing $\mathfrak{c}$ with dual vector kinematics that gauge invariance of
the overall amplitude is preserved. Color structures like $d^{4}f^{3}$ demand
dual gauge theory solutions; we have not yet identified consistently
factorizing $d^{4}f^{3}$-dual vector numerators. Further exploration of gauge
theory solutions could yield additional gravity solutions of interest. We
additionally find that there is a second avenue for generating local gravity
corrections from our doubly dual amplitudes by dressing each graph with two
copies of $\mathfrak{f}_{\mathfrak{c}}(n^{\text{vec}}_{\mathfrak{a}},s_{ij})$,
which is guaranteed to be gauge invariant.

We verify our proposition that these results — both factorizing and local —
span the low-energy expansion of the five point open superstring amplitude
through $\mathcal{O}\left({\alpha^{\prime}}^{9}\right)$. We note that we find
additional local solutions to those appearing in the open superstring at tree-
level, both for color structures compatible with string theory
($f^{3}f^{3}f^{3}$ and $d^{4}f^{3}$) and those forbidden by monodromy
($f^{3}d^{3}f^{3}$ and $d^{5}$).

When we restrict ourselves to manifestly bi-adjoint higher-derivative color-
weights, we suggest a procedure for constructing a modified KLT kernel, eq.
(183), which encodes all higher-derivative corrections compatible with two
adjoint-type algebraic structures. Our approach of summing over all double-
copies of algebraic structures given in terms of adjoint-color-weights and
momentum invariants in eq. (180) guarantees that there can be no missing form
of higher-derivative correction, but it does not guarantee a lack of
redundancy. Indeed, the existence of casting between e.g. $f^{3}f^{3}f^{3}$
and $d^{4}f^{3}$ algebraic structures suggests a potentially large amount of
redundancy. This leaves us with a number of interesting open questions that
would be exciting to pursue: Is there an efficient way of isolating the unique
contributions? Is there an analogy to the unit-step linear relaxed numerator
that can be composed with KLT to generate all KLT-type higher-derivative
corrections?

We have not provided explicit operators for any of the higher-derivative
corrections we present here, save a pedagogical example in Appendix C. We note
that identifying operators associated with particular predictions can be
accomplished by giving an appropriate mass-dimension ansatz to an action and
constraining it on Feynman rules. Yet this is both laborious and tedious, and
only grows more so as multiplicity and higher-mass-dimension increases. While
arguably the gauge-invariant prediction is ultimately what matters, we find
this situation limiting. It would be interesting to see if Hilbert-series
methods, see e.g. Henning:2015alf ; Lehman:2015via ; Lehman:2015coa ;
Henning:2015daa ; Henning:2017fpj , can be applied to make constructive
double-copy structure manifest. We expect that any approach towards
identifying efficient means of generating higher-derivative operators relevant
to these higher-derivative predictions will also aid in clarifying open
questions regarding double-copy construction and the web-of-theories.

The fact that the encoding of higher derivative supersymmetry-compatible
corrections to gauge theory and gravity within scalar theories, constructed
from simple color and scalar kinematic building blocks and composition rules,
continues at five points indicates the potential for using these techniques at
both higher multiplicity and at loop level. In particular, the five-point
results of this paper could be used through unitarity to construct an infinite
tower of higher derivative multiloop corrections, as unitarity cuts of two
loop integrands may be fixed on the five point tree level results presented
here and in the auxiliary files ancFiles . This method shows promise at loop
level, since capturing all higher derivative corrections in modified color
factors would allow us to trivially recycle hard-won multiloop gauge theory
integrand numerators into their higher derivative corrections:

$\mathcal{A}_{L-\text{loops}}^{\text{sYM+HD}}=\sum_{g}\frac{1}{S_{g}}\int\prod_{i}^{L}\frac{d^{D}l_{i}}{(2\pi)^{D}}\,\,\frac{n^{\text{sYM}}_{L}(g)c^{\text{HD}}_{L}(g)}{d(g)}$
(207)

Adjoint-dual loop level integrand numerators $n^{\text{sYM}}_{L}$ have been
found through four loops SimplifyingBCJ . We conjecture that, for suitable
integrand numerators $c^{\text{HD}}_{L}$, colors factors could be
appropriately replaced to generate higher derivative gravity integrands via
double copy.

There are also many opportunities for further work at tree level. At four
points, we found that only four tensor structures in addition to Yang-Mills
were necessary to generate all corrections within the bosonic open string;
non-supersymmetric vector numerators at five points that could yield more
general corrections remain to be explored. Of further interest are other color
algebraic structures: in particular, multitrace amplitudes and predictions
concerning states charged in the fundamental were recently shown to be
expressible in terms of modified color factors for the NLSM and higher
derivatives Low:2019wuv ; Low:2020ubn . In addition to expanding this
framework to be compatible with fundamental color-kinematics duality,
extending it to include massive particles could allow for the description of
corrections to QCD containing standard model fermions, which should be of
interest in the context of standard model effective field theory.

###### Acknowledgements.

We are grateful to Yu-tin Huang, Henrik Johansson, Ian Low, Nic Pavao, Radu
Roiban, Oliver Schlotterer, Aslan Seifi, Bogdan Stoica, and Ingrid Vazquez-
Holm for interesting discussion of related subjects. We are additionally
grateful for especially helpful comments by Oliver Schlotterer on an earlier
draft. This work is supported by the Department of Energy under Award Number
DE-SC0021485. LR is supported by Taiwan Ministry of Science and Technology
Grant No. 109-2811-M-002-523. SZ would like to acknowledge the Northwestern
University Amplitudes and Insight group, Department of Physics and Astronomy,
and Weinberg College for support.

##  Appendix A Finding Composition Rules

At four points, we may generate a new adjoint numerator $n$ from the
composition of two known adjoint numerators $j$ and $k$:

$n_{\mathfrak{a}}(1234)=\left(j_{\mathfrak{a}}\,$\mathfrak{a}$⃝\,k_{\mathfrak{a}}\right)=j_{\mathfrak{a}}(4123)k_{\mathfrak{a}}(4123)-j_{\mathfrak{a}}(4231)k_{\mathfrak{a}}(4231)$
(208)

Although it is not difficult to verify that this form ensures that
$n_{\mathfrak{a}}$ satisfies antisymmetry and Jacobi, we will construct it
explicitly here, so it is clear how more complicated compositions may be
generated at five points.

To begin, let us make a judicious choice of basis for four point graphs,
$g(4123)$ and $g(4231)$, corresponding to the $t$ and $u$ channel graphs
respectively. We start by writing down an ansatz for the composed numerator
$n$ as the outer product of these basis sets for numerators $j_{\mathfrak{a}}$
and $k_{\mathfrak{a}}$:

$\begin{split}n_{\mathfrak{a}}(1234)=\alpha
j_{\mathfrak{a}}(4123)k_{\mathfrak{a}}(4123)+\beta
j_{\mathfrak{a}}(4231)k_{\mathfrak{a}}(4123)\\\ +\gamma
j_{\mathfrak{a}}(4123)k_{\mathfrak{a}}(4231)+\delta
j_{\mathfrak{a}}(4231)k_{\mathfrak{a}}(4231)\end{split}$ (209)

First, impose antisymmetry on $n_{\mathfrak{a}}$, using the antisymmetry of
$j_{\mathfrak{a}}$ and $k_{\mathfrak{a}}$ to return to the basis and constrain
the ansatz:

$\begin{split}n_{\mathfrak{a}}(2134)=\alpha
j_{\mathfrak{a}}(4213)k_{\mathfrak{a}}(4213)+\beta
j_{\mathfrak{a}}(4132)k_{\mathfrak{a}}(4213)\\\ +\gamma
j_{\mathfrak{a}}(4213)k_{\mathfrak{a}}(4132)+\delta
j_{\mathfrak{a}}(4132)k_{\mathfrak{a}}(4132)\end{split}$ (210)
$\begin{split}n_{\mathfrak{a}}(2134)=\alpha
j_{\mathfrak{a}}(4231)k_{\mathfrak{a}}(4231)+\beta
j_{\mathfrak{a}}(4123)k_{\mathfrak{a}}(4231)\\\ +\gamma
j_{\mathfrak{a}}(4231)k_{\mathfrak{a}}(4123)+\delta
j_{\mathfrak{a}}(4123)k_{\mathfrak{a}}(4123)\end{split}$ (211)
$\begin{split}n_{\mathfrak{a}}(2134)+n_{\mathfrak{a}}(1234)=(\alpha+\delta)j_{\mathfrak{a}}(4231)k_{\mathfrak{a}}(4231)+(\beta+\gamma)j_{\mathfrak{a}}(4123)k_{\mathfrak{a}}(4231)\\\
+(\beta+\gamma)j_{\mathfrak{a}}(4231)k_{\mathfrak{a}}(4123)+(\beta+\gamma)j_{\mathfrak{a}}(4123)k_{\mathfrak{a}}(4123)\end{split}$
(212)
$n_{\mathfrak{a}}(2134)+n_{\mathfrak{a}}(1234)=0\implies\delta=-\alpha,\quad\gamma=-\beta$
(213)

Thus, antisymmetry reduces our ansatz down to two parameters:

$\begin{split}n_{\mathfrak{a}}(1234)=\alpha[j_{\mathfrak{a}}(4123)k_{\mathfrak{a}}(4123)-j_{\mathfrak{a}}(4231)k_{\mathfrak{a}}(4231)]\\\
+\beta[j_{\mathfrak{a}}(4231)k_{\mathfrak{a}}(4123)-j_{\mathfrak{a}}(4123)k_{\mathfrak{a}}(4231)]\end{split}$
(214)

Now we must impose Jacobi:

$n_{\mathfrak{a}}(1234)=n_{\mathfrak{a}}(4123)+n_{\mathfrak{a}}(4231)$ (215)
$\begin{split}\alpha[j_{\mathfrak{a}}(4123)k_{\mathfrak{a}}(4123)-j_{\mathfrak{a}}(4231)k_{\mathfrak{a}}(4231)]+\beta[j_{\mathfrak{a}}(4231)k_{\mathfrak{a}}(4123)-j_{\mathfrak{a}}(4123)k_{\mathfrak{a}}(4231)]\\\
=\alpha[j_{\mathfrak{a}}(1234)k_{\mathfrak{a}}(1234)-j_{\mathfrak{a}}(4231)k_{\mathfrak{a}}(4231)]+\beta[-j_{\mathfrak{a}}(4231)k_{\mathfrak{a}}(1234)+j_{\mathfrak{a}}(1234)k_{\mathfrak{a}}(4231)]\\\
+\alpha[j_{\mathfrak{a}}(4123)k_{\mathfrak{a}}(4123)-j_{\mathfrak{a}}(1234)k_{\mathfrak{a}}(1234)]+\beta[-j_{\mathfrak{a}}(1234)k_{\mathfrak{a}}(4123)+j_{\mathfrak{a}}(4123)k_{\mathfrak{a}}(1234)]\end{split}$
(216)
$\begin{split}0=\beta[j_{\mathfrak{a}}(4231)k_{\mathfrak{a}}(4123)-j_{\mathfrak{a}}(4123)k_{\mathfrak{a}}(4231)+j_{\mathfrak{a}}(4231)k_{\mathfrak{a}}(1234)\\\
-j_{\mathfrak{a}}(1234)k_{\mathfrak{a}}(4231)+j_{\mathfrak{a}}(1234)k_{\mathfrak{a}}(4123)-j_{\mathfrak{a}}(4123)k_{\mathfrak{a}}(1234)]\end{split}$
(217)

Using the fact that both $j_{\mathfrak{a}}$ and $k_{\mathfrak{a}}$ themselves
obey Jacobi allows us to express this relation in terms of basis elements, and
we find $\beta=0$, leaving us with the simple composition rule:

$n_{\mathfrak{a}}(1234)=j_{\mathfrak{a}}(4123)k_{\mathfrak{a}}(4123)-j_{\mathfrak{a}}(4231)k_{\mathfrak{a}}(4231)$
(218)

We proceed to find compositions at multiplicity five in a similar manner. The
general form of a composition rule that takes two numerators of given
algebraic structures $\mathfrak{e}$ and $\mathfrak{f}$ and composes them into
a new numerator desired algebraic structure $\mathfrak{g}$ is:

$\mathfrak{g}(12345)=\left({\mathfrak{e}}\,\small$\mathfrak{g}$⃝\,{\mathfrak{f}}\right)=\sum_{g\in\rho_{\mathfrak{e}}}\sum_{g^{\prime}\in\rho_{\mathfrak{f}}}\beta_{g,g^{\prime}}^{\mathfrak{efg}}\mathfrak{e}(g)\mathfrak{f}(g^{\prime})$
(219)

where the sums run over the basis graphs $\rho$ for the given type of
numerator. The coefficients
$\beta_{g,g^{\prime}}^{\mathfrak{f},\mathfrak{g},\mathfrak{e}}$ are chosen
such that $n_{\mathfrak{e}}$ obeys the $\mathfrak{e}$-type algebraic
constraints, simply by virtue of the algebraic properties of
$j_{\mathfrak{f}}$ and $k_{\mathfrak{g}}$.

##  Appendix B Color Structure Basis at Five-Points

We choose our 24 element basis of color structures at five points (in
agreement with the $4!$ trace basis elements) to be given as follows: first,
there is the single permutation invariant structure $d^{5}$. Then,
antisymmetry and Jacobi constrain us to a basis of six adjoint structures
$f^{3}f^{3}f^{3}$; we choose the $(m-2)!$ DDM basis $f^{3}f^{3}f^{3}(1\sigma
5)$:

$\begin{split}f^{3}f^{3}f^{3}(12345)\quad f^{3}f^{3}f^{3}(12435)\quad
f^{3}f^{3}f^{3}(13245)\\\ f^{3}f^{3}f^{3}(13425)\quad
f^{3}f^{3}f^{3}(14235)\quad f^{3}f^{3}f^{3}(14325)\end{split}$ (220)

Based on the hybrid algebraic constraints of eqs. (94) and (96), we land on a
basis of six $d^{4}f^{3}$ color structures:

$\begin{split}&d^{4}f^{3}(12345)\quad d^{4}f^{3}(12435)\quad
d^{4}f^{3}(12534)\\\ &d^{4}f^{3}(14523)\quad d^{4}f^{3}(23415)\quad
d^{4}f^{3}(23514)\\\ \end{split}$ (221)

Finally, the sandwich constraints given in eqs. (88) and (91) give an eleven-
element basis of $f^{3}d^{3}f^{3}$ color structures:

$\begin{split}&f^{3}d^{3}f^{3}(12345)\quad f^{3}d^{3}f^{3}(12435)\quad
f^{3}d^{3}f^{3}(12534)\\\ &f^{3}d^{3}f^{3}(13245)\quad
f^{3}d^{3}f^{3}(13425)\quad f^{3}d^{3}f^{3}(13524)\\\
&f^{3}d^{3}f^{3}(14235)\quad f^{3}d^{3}f^{3}(14325)\quad
f^{3}d^{3}f^{3}(14523)\\\ &f^{3}d^{3}f^{3}(23145)\quad
f^{3}d^{3}f^{3}(24135)\quad\end{split}$ (222)

We have made the choice to prioritize $f^{3}d^{3}f^{3}$ structures in our
basis over $d^{3}f^{3}f^{3}$; these structures may be expressed completely in
terms of the $f^{3}d^{3}f^{3}$ structures as follows:

$d^{3}f^{3}f^{3}(abcde)=f^{3}d^{3}f^{3}(acbde)+f^{3}d^{3}f^{3}(bcade)\,.$
(223)

##  Appendix C Operator Comparison at Five-Points

In this appendix, we briefly discuss an example of lining up our amplitudes
with results from a traditional operator approach using Feynman rules. We will
use as our example the lowest order higher-derivative correction to super-
Yang-Mills theory at five points, which is a factorizing correction carrying
$d^{4}f^{3}$ color at second order in ${\alpha^{\prime}}$. The relevant action
for this correction is given by:

$\mathcal{S}=\int
d^{d}x\left(\mathcal{L}_{\text{YM}}+{\alpha^{\prime}}^{2}F^{4}_{\text{SUSY}}\right)\,$
(224)

where the operators are defined as follows:

$\mathcal{L}_{\text{YM}}\equiv-\frac{1}{4}\operatorname{Tr}\left[F_{\mu\nu}F^{\mu\nu}\right]\,$
(225)
$F^{4}_{\text{SUSY}}\equiv\operatorname{Tr}\left[F_{\mu}^{\nu}F_{\nu}^{\rho}F_{\rho}^{\sigma}F_{\sigma}^{\mu}+2F_{\mu}^{\nu}F_{\rho}^{\sigma}F_{\nu}^{\rho}F_{\sigma}^{\mu}-\tfrac{1}{4}F_{\mu\nu}F_{\rho\sigma}F^{\mu\nu}F^{\rho\sigma}-\tfrac{1}{2}F_{\mu\nu}F^{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}\right]\,.$
(226)
$F_{\mu\nu}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+\frac{i}{\sqrt{2}}g\left(A_{\mu}A_{\nu}-A_{\nu}A_{\mu}\right)\,.$
(227)

The Feynman rules calculation for this five-point correction contains two
parts: one five-point contact contribution arising from the five-field terms
within $F^{4}_{\text{SUSY}}$, and one factorizing contribution arising from
the four-field terms of $F^{4}_{\text{SUSY}}$ sewn together with a three-point
Yang-Mills insertion. Schematically, with $F^{4}$-vertices drawn with filled
(teal) circles and standard Yang-Mills vertices as open (pink) circles,

$\mathcal{A}_{\text{action}}={\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex[dot, ultra thick, teal] (mid2) at (0,0) {}; \vertex(a1) at
(-0.58778525229, -0.80901699437) {$$}; \vertex(a2) at
(-0.9510565,0.30901699437) {$$}; \vertex(a3) at (0,1) {$$}; \vertex(a4) at
(0.9510565,0.30901699437) {$$}; \vertex(a5) at (0.58778525229, -0.80901699437)
{$$}; \diagram{ (mid2)--[thick](a1), (mid2)--[thick](a2), (mid2)--[thick](a3),
(mid2)--[thick](a4), (mid2)--[thick](a5) }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+\sum_{g\in\Gamma_{\mathfrak{h}}}^{10}{\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a1) at (-.1, -0.9) {$$}; \vertex(a2) at (-1, 0) {$$}; \vertex[dot,
ultra thick, teal] (mid2) at (-0.1,0) {}; \vertex[empty dot, ultra thick,
magenta] (mid3) at (1.2, 0) {}; \vertex(a5) at (-.1, 0.9) {$$}; \vertex(a3) at
(1.2, 0.9) {$$}; \vertex(a4) at (1.2, -0.9) {$$}; \diagram{ (mid2)
--[thick](a5), (mid2) -- [thick](mid3), (a1)--[thick](mid2), (a2) --
[thick](mid2), (mid3)--[thick](a3), (a4) -- [thick](mid3) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ (228)

The factorizing contribution must be found by dressing the ten hybrid graphs
$\Gamma_{\mathfrak{h}}$ at five points: the quartic vertex is dressed with the
off-shell four-point vertex from the $F^{4}_{\text{SUSY}}$ operator, the cubic
vertex is dressed with the off-shell three-point Yang-Mills vertex, and each
graph is dressed with its associated propagator. The contact contribution
dresses the single quintic graph at five points. Both of these contributions
are necessary for gauge invariance of the overall amplitude.

Our method for constructing this correction involves an adjoint double copy
between Yang Mills and our higher-derivative bicolor theory: each cubic graph
is dressed with an adjoint Yang-Mills numerator and the second order adjoint
modified color weight:

$\mathcal{A}_{\text{double-
copy}}=\sum_{g\in\Gamma_{3}}^{15}\frac{n^{\text{YM}}_{g}c^{\text{HD(2)}}_{g}}{d_{g}}\,.$
(229)

The explicit form of this higher-derivative color weight, generated via
compositions fixed to factorize appropriately down to four-point higher-
derivative corrections times the three-point uncorrected bi-adjoint scalar, is
given in the ancillary files and reproduced here:

$\begin{split}c^{\text{HD(2)}}(12345)&=\frac{2}{5}\,b[2,1]\,\Big{(}d^{4}f^{3}(14523)s_{45}\left(-s_{12}+2s_{34}+s_{45}\right)\\\
&+d^{4}f^{3}(12345)\left(s_{12}^{2}+s_{12}(9s_{23}-6s_{45})-s_{45}(5s_{23}+s_{45})\right)\\\
&+d^{4}f^{3}(12435)\left((5s_{15}-5s_{23}-8s_{34}-2s_{45})s_{45}+s_{12}(5s_{15}+s_{23}+4s_{45})\right)\\\
&+d^{4}f^{3}(23514)\left(s_{12}^{2}-s_{45}(9s_{34}+s_{45})+s_{12}(5s_{34}+6s_{45})\right)\\\
&+d^{4}f^{3}(23415)\left(-6s_{12}^{2}+s_{12}(5s_{34}+8s_{45})-s_{45}(9s_{34}+8s_{45})\right)\\\
&+d^{4}f^{3}(12534)\big{(}(5s_{15}-8s_{34}-s_{45})s_{45}+s_{12}(5s_{15}-6s_{23}+9s_{45})\big{)}\Big{)}\,.\end{split}$
(230)

We find that fixing $b[2,1]=6\sqrt{2}\,i\,g$ allows for agreement between our
amplitude calculation and the Feynman rules calculation:

$\mathcal{A}_{\text{double-copy}}=\mathcal{A}_{\text{action}}\,,$ (231)

verifying the compatibility of our double copy construction method with
higher-derivative color weights with traditional operator calculations.

##  Appendix D Relevant Composition at Five-Points

$\displaystyle\left(j_{\mathfrak{r}}\,$\mathfrak{r}$⃝\,k_{\mathfrak{r}}\right)=-2j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(13245)\
-2j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(12435)\
-2j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(13425)$ $\displaystyle\
-2j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(14235)\
-2j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(13425)\
-2j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(14235)$ $\displaystyle\
+3j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(13425)\
+3j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(14235)\
+3j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(13425)$ $\displaystyle\
+3j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(14235)\
+3j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(12435)\
+3j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(13245)$ $\displaystyle\
+3j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(12435)\
+3j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(13245)\
-4j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(12435)$ $\displaystyle\
-4j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(13245)\
+4j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(14325)\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(14325)$ $\displaystyle\
-j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(14325)\
-j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(13425)\
-j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(14235)$

$\displaystyle\left(j_{\mathfrak{r}}\,$\mathfrak{a}$⃝\,k_{\mathfrak{r}}\right)=-2j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(14325)\
+2j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(14325)\
+2j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(12435)$ $\displaystyle\
-2j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(13245)\
+3j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(14325)\
-3j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(14325)$ $\displaystyle\
-3j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(13425)\
+3j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(14235)\
+5j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(14235)$ $\displaystyle\
-5j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(13425)\
+5j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(13245)\
-5j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(12435)$ $\displaystyle\
-6j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(14235)\
+6j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(13425)\
-j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(13425)$ $\displaystyle\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(14235)\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(12435)\
-j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(13245)$

$\displaystyle\left(j_{\mathfrak{r}}\,$\mathfrak{a}$⃝\,k_{\mathfrak{a}}\right)=a_{1}\Big{(}2j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(13425)\
+2j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(14235)$ $\displaystyle\
+2j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(12435)\
+2j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(13245)$ $\displaystyle\
+2j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(12435)\
+2j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(13245)$ $\displaystyle\
-2j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(12435)\
-2j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(13245)$ $\displaystyle\
-3j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(12345)\
-3j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(12345)$ $\displaystyle\
+4j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(12345)\
-j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(12435)$ $\displaystyle\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(13245)\
-j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(14325)$ $\displaystyle\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(12435)\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(13245)$ $\displaystyle\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(14325)\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(12345)$ $\displaystyle\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(13425)\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(14235)$ $\displaystyle\
+j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(12345)\
-j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(13425)$ $\displaystyle\
-j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(14235)\
+j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(13425)$ $\displaystyle\
+j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(14235)\Big{)}$
$\displaystyle+a_{2}\Big{(}\ 2j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(12345)\
-j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(12345)$ $\displaystyle\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(13425)\
-j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(14235)$ $\displaystyle\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(12345)\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(13425)$ $\displaystyle\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(14235)\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(12435)$ $\displaystyle\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(13245)\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(14325)$ $\displaystyle\
+j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(12435)\
+j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(13245)$ $\displaystyle\
-j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(14325)\
-j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(12435)$ $\displaystyle\
-j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(13245)\
+j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(14325)\Big{)}$

$\displaystyle\left(j_{\mathfrak{r}}\,$\mathfrak{h}$⃝\,k_{\mathfrak{r}}\right)=-2j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(14235)\
+2j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(14325)\
+2j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(12435)$ $\displaystyle\
-2j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(12435)\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(13245)\
-j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(13425)$ $\displaystyle\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(12435)\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(13425)\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(12435)$ $\displaystyle\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(13245)\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(14235)\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(14325)$ $\displaystyle\
-j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(13425)\
+j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(13425)$

$\displaystyle\left(j_{\mathfrak{r}}\,$\mathfrak{h}$⃝\,k_{\mathfrak{h}}\right)=a_{1}\Big{(}2j_{\mathfrak{r}}(14235)k_{\mathfrak{h}}(13524)\
-j_{\mathfrak{r}}(12435)k_{\mathfrak{h}}(12534)$ $\displaystyle\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{h}}(14523)\
-j_{\mathfrak{r}}(12435)k_{\mathfrak{h}}(23514)$ $\displaystyle\
-j_{\mathfrak{r}}(12435)k_{\mathfrak{h}}(34512)\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{h}}(12534)$ $\displaystyle\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{h}}(13524)\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{h}}(12534)$ $\displaystyle\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{h}}(14523)\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{h}}(23514)$ $\displaystyle\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{h}}(24513)\
+j_{\mathfrak{r}}(14235)k_{\mathfrak{h}}(12534)$ $\displaystyle\
+j_{\mathfrak{r}}(14235)k_{\mathfrak{h}}(23514)\
+j_{\mathfrak{r}}(14325)k_{\mathfrak{h}}(12534)$ $\displaystyle\
+j_{\mathfrak{r}}(14325)k_{\mathfrak{h}}(23514)\Big{)}$
$\displaystyle+a_{2}\Big{(}\ 2j_{\mathfrak{r}}(13245)k_{\mathfrak{h}}(24513)\
+2j_{\mathfrak{r}}(13245)k_{\mathfrak{h}}(34512)$ $\displaystyle\
-2j_{\mathfrak{r}}(13425)k_{\mathfrak{h}}(34512)\
+2j_{\mathfrak{r}}(14235)k_{\mathfrak{h}}(14523)$ $\displaystyle\
-2j_{\mathfrak{r}}(14235)k_{\mathfrak{h}}(34512)\
-2j_{\mathfrak{r}}(14325)k_{\mathfrak{h}}(14523)$ $\displaystyle\
+2j_{\mathfrak{r}}(14325)k_{\mathfrak{h}}(34512)\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{h}}(13524)$ $\displaystyle\
-j_{\mathfrak{r}}(12435)k_{\mathfrak{h}}(14523)\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{h}}(34512)$ $\displaystyle\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{h}}(23514)\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{h}}(12534)$ $\displaystyle\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{h}}(14523)\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{h}}(23514)$ $\displaystyle\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{h}}(24513)\
-j_{\mathfrak{r}}(14235)k_{\mathfrak{h}}(12534)$ $\displaystyle\
-j_{\mathfrak{r}}(14235)k_{\mathfrak{h}}(23514)\
-j_{\mathfrak{r}}(14325)k_{\mathfrak{h}}(13524)\Big{)}$

$\displaystyle\left(j_{\mathfrak{r}}\,$\mathfrak{s}$⃝\,k_{\mathfrak{r}}\right)=a_{1}\Big{(}-2j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(13245)\
-2j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(12435)$ $\displaystyle\
-2j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(13425)\
-2j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(14235)$ $\displaystyle\
-2j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(13425)\
-2j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(14235)$ $\displaystyle\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(13425)\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(14235)$ $\displaystyle\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(13425)\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(14235)$ $\displaystyle\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(12435)\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(13245)$ $\displaystyle\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(14325)\
+j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(12435)$ $\displaystyle\
+j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(13245)\
+j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(14325)$ $\displaystyle\
+j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(13425)\
+j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(14235)\Big{)}$
$\displaystyle+a_{2}\Big{(}\ -j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(12435)\
-j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(13245)$ $\displaystyle\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(13425)\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(14235)$ $\displaystyle\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(12435)\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(13245)$ $\displaystyle\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(13425)\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(14235)$ $\displaystyle\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(12435)\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(13245)$ $\displaystyle\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(13425)\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(14235)$ $\displaystyle\
+j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(12435)\
+j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(13245)$ $\displaystyle\
-j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(13425)\
-j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(14235)$ $\displaystyle\
+j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(14325)\Big{)}$

$\displaystyle\left(j_{\mathfrak{r}}\,$\mathfrak{h}$⃝\,k_{\mathfrak{a}}\right)=a_{1}\Big{(}-2j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(12345)\
-2j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(14325)$ $\displaystyle\
-2j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(13425)\
+2j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(12345)$ $\displaystyle\
-2j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(12435)\
+2j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(14235)$ $\displaystyle\
-3j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(13245)\
+3j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(14325)$ $\displaystyle\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(13245)\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(13425)$ $\displaystyle\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(12345)\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(13245)$ $\displaystyle\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(14235)\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(14325)$ $\displaystyle\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(12435)\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(14235)$ $\displaystyle\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(14325)\
-j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(12345)$ $\displaystyle\
-j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(12435)\
+j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(13245)$ $\displaystyle\
+j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(14235)\
-j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(13425)\Big{)}$
$\displaystyle+a_{2}\Big{(}\ 2j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(12435)\
-2j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(14235)$ $\displaystyle\
+2j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(12435)\
+2j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(13425)$ $\displaystyle\
-2j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(13425)\
-2j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(13245)$ $\displaystyle\
+2j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(14325)\
-3j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(12345)$ $\displaystyle\
-3j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(12435)\
+3j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(13245)$ $\displaystyle\
+3j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(14235)\
-4j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(12345)$ $\displaystyle\
+4j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(12345)\
-5j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(13245)$ $\displaystyle\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(13245)\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(13425)$ $\displaystyle\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(12345)\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(13245)$ $\displaystyle\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(14235)\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(14325)$ $\displaystyle\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(12435)\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(14235)$ $\displaystyle\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(14325)\
+j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(14325)$ $\displaystyle\
-j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(13425)\Big{)}$

$\displaystyle\left({j_{\mathfrak{r}}}\,\small$\mathfrak{p}$⃝\,{k_{\mathfrak{r}}}\right)=-2j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(13425)\
-2j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(14235)\
-2j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(13425)$ $\displaystyle\
-2j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(14235)\
-2j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(12435)\
-2j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(13245)$ $\displaystyle\
+2j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(14235)\
-2j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(14325)\
-2j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(12435)$ $\displaystyle\
-2j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(13245)\
+2j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(13425)\
-2j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(14325)$ $\displaystyle\
-2j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(13425)\
-2j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(14235)\
+3j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(12435)$ $\displaystyle\
+3j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(13245)\
+3j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(14325)\
+4j_{\mathfrak{r}}(13425)k_{\mathfrak{r}}(13425)$ $\displaystyle\
+4j_{\mathfrak{r}}(14235)k_{\mathfrak{r}}(14235)\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(13245)\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{r}}(14325)$ $\displaystyle\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(12435)\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{r}}(14325)\
+j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(12435)$ $\displaystyle\
+j_{\mathfrak{r}}(14325)k_{\mathfrak{r}}(13245)$

$\displaystyle\left({j_{\mathfrak{a}}}\,\small$\mathfrak{p}$⃝\,{k_{\mathfrak{a}}}\right)=-2j_{\mathfrak{a}}(12345)k_{\mathfrak{a}}(12435)\
-2j_{\mathfrak{a}}(12345)k_{\mathfrak{a}}(13245)\
+2j_{\mathfrak{a}}(12345)k_{\mathfrak{a}}(14325)$ $\displaystyle\
-2j_{\mathfrak{a}}(12435)k_{\mathfrak{a}}(12345)\
+2j_{\mathfrak{a}}(12435)k_{\mathfrak{a}}(13425)\
-2j_{\mathfrak{a}}(12435)k_{\mathfrak{a}}(14235)$ $\displaystyle\
-2j_{\mathfrak{a}}(13245)k_{\mathfrak{a}}(12345)\
-2j_{\mathfrak{a}}(13245)k_{\mathfrak{a}}(13425)\
+2j_{\mathfrak{a}}(13245)k_{\mathfrak{a}}(14235)$ $\displaystyle\
+2j_{\mathfrak{a}}(13425)k_{\mathfrak{a}}(12435)\
-2j_{\mathfrak{a}}(13425)k_{\mathfrak{a}}(13245)\
-2j_{\mathfrak{a}}(13425)k_{\mathfrak{a}}(14325)$ $\displaystyle\
-2j_{\mathfrak{a}}(14235)k_{\mathfrak{a}}(12435)\
+2j_{\mathfrak{a}}(14235)k_{\mathfrak{a}}(13245)\
-2j_{\mathfrak{a}}(14235)k_{\mathfrak{a}}(14325)$ $\displaystyle\
+2j_{\mathfrak{a}}(14325)k_{\mathfrak{a}}(12345)\
-2j_{\mathfrak{a}}(14325)k_{\mathfrak{a}}(13425)\
-2j_{\mathfrak{a}}(14325)k_{\mathfrak{a}}(14235)$ $\displaystyle\
+5j_{\mathfrak{a}}(12345)k_{\mathfrak{a}}(12345)\
+5j_{\mathfrak{a}}(12435)k_{\mathfrak{a}}(12435)\
+5j_{\mathfrak{a}}(13245)k_{\mathfrak{a}}(13245)$ $\displaystyle\
+5j_{\mathfrak{a}}(13425)k_{\mathfrak{a}}(13425)\
+5j_{\mathfrak{a}}(14235)k_{\mathfrak{a}}(14235)\
+5j_{\mathfrak{a}}(14325)k_{\mathfrak{a}}(14325)$ $\displaystyle\
-j_{\mathfrak{a}}(12345)k_{\mathfrak{a}}(13425)\
-j_{\mathfrak{a}}(12345)k_{\mathfrak{a}}(14235)\
-j_{\mathfrak{a}}(12435)k_{\mathfrak{a}}(13245)$ $\displaystyle\
-j_{\mathfrak{a}}(12435)k_{\mathfrak{a}}(14325)\
-j_{\mathfrak{a}}(13245)k_{\mathfrak{a}}(12435)\
-j_{\mathfrak{a}}(13245)k_{\mathfrak{a}}(14325)$ $\displaystyle\
-j_{\mathfrak{a}}(13425)k_{\mathfrak{a}}(12345)\
-j_{\mathfrak{a}}(13425)k_{\mathfrak{a}}(14235)\
-j_{\mathfrak{a}}(14235)k_{\mathfrak{a}}(12345)$ $\displaystyle\
-j_{\mathfrak{a}}(14235)k_{\mathfrak{a}}(13425)\
-j_{\mathfrak{a}}(14325)k_{\mathfrak{a}}(12435)\
-j_{\mathfrak{a}}(14325)k_{\mathfrak{a}}(13245)$

$\displaystyle\left(j_{\mathfrak{r}}\,$\mathfrak{s}$⃝\,k_{\mathfrak{a}}\right)=a_{1}\Big{(}j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(12345)\
-j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(12435)$ $\displaystyle\
-j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(13245)\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(12345)$ $\displaystyle\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(12435)\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(13245)$ $\displaystyle\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(12345)\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(12435)$ $\displaystyle\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(13245)\
-j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(12345)$ $\displaystyle\
+j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(12435)\
-j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(13245)$ $\displaystyle\
-j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(12435)\
+j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(13245)\Big{)}$
$\displaystyle+a_{2}\Big{(}\ -j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(12345)\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(12435)$ $\displaystyle\
+j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(13245)\
-j_{\mathfrak{r}}(12435)k_{\mathfrak{a}}(14325)$ $\displaystyle\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(12345)\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(12435)$ $\displaystyle\
-j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(13245)\
+j_{\mathfrak{r}}(13245)k_{\mathfrak{a}}(14325)$ $\displaystyle\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(12345)\
+j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(13425)$ $\displaystyle\
-j_{\mathfrak{r}}(13425)k_{\mathfrak{a}}(14235)\
-j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(12345)$ $\displaystyle\
+j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(13425)\
-j_{\mathfrak{r}}(14235)k_{\mathfrak{a}}(14235)$ $\displaystyle\
-j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(13425)\
+j_{\mathfrak{r}}(14325)k_{\mathfrak{a}}(14235)\Big{)}$

$\displaystyle\left(j_{\mathfrak{s}}\,$\mathfrak{s}$⃝\,k_{\mathfrak{a}}\right)=a_{1}\Big{(}3j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(13245)\
-3j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(14235)$ $\displaystyle\
-3j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(12345)\
+3j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(13245)$ $\displaystyle\
+3j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(13425)\
-3j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(14325)$ $\displaystyle\
-3j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(12435)\
+3j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(13425)$ $\displaystyle\
+3j_{\mathfrak{s}}(23145)k_{\mathfrak{a}}(12345)\
-3j_{\mathfrak{s}}(23145)k_{\mathfrak{a}}(12435)$ $\displaystyle\
-3j_{\mathfrak{s}}(23145)k_{\mathfrak{a}}(14235)\
+3j_{\mathfrak{s}}(23145)k_{\mathfrak{a}}(14325)$ $\displaystyle\
-j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(12345)\
+j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(12435)$ $\displaystyle\
-j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(13425)\
+j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(14325)$ $\displaystyle\
+j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(12435)\
+j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(14235)$ $\displaystyle\
+j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(12345)\
-j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(13245)$ $\displaystyle\
+j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(14235)\
-j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(14325)$ $\displaystyle\
-j_{\mathfrak{s}}(23145)k_{\mathfrak{a}}(13245)\
-j_{\mathfrak{s}}(23145)k_{\mathfrak{a}}(13425)\Big{)}$
$\displaystyle+a_{2}\Big{(}\ -2j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(13425)\
+2j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(14325)$ $\displaystyle\
+2j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(14235)\
+2j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(14235)$ $\displaystyle\
-2j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(14325)\
-2j_{\mathfrak{s}}(23145)k_{\mathfrak{a}}(13425)$ $\displaystyle\
+3j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(13245)\
+3j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(13245)$ $\displaystyle\
-3j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(12435)\
-3j_{\mathfrak{s}}(13425)k_{\mathfrak{a}}(12345)$ $\displaystyle\
-3j_{\mathfrak{s}}(13524)k_{\mathfrak{a}}(12345)\
+3j_{\mathfrak{s}}(14235)k_{\mathfrak{a}}(12345)$ $\displaystyle\
-3j_{\mathfrak{s}}(23145)k_{\mathfrak{a}}(12435)\
+3j_{\mathfrak{s}}(24135)k_{\mathfrak{a}}(12345)$ $\displaystyle\
+j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(12345)\
-j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(12435)$ $\displaystyle\
-j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(12435)\
-j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(12345)$ $\displaystyle\
+j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(13245)\
+j_{\mathfrak{s}}(23145)k_{\mathfrak{a}}(13245)\Big{)}$
$\displaystyle+a_{3}\Big{(}\ j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(12345)\
-j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(12435)$ $\displaystyle\
-j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(13425)\
+j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(14325)$ $\displaystyle\
-j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(12435)\
+j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(13245)$ $\displaystyle\
+j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(14235)\
-j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(12345)$ $\displaystyle\
+j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(13245)\
+j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(14235)$ $\displaystyle\
-j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(14325)\
-j_{\mathfrak{s}}(13425)k_{\mathfrak{a}}(12345)$ $\displaystyle\
+j_{\mathfrak{s}}(13425)k_{\mathfrak{a}}(12435)\
-j_{\mathfrak{s}}(13524)k_{\mathfrak{a}}(13245)$ $\displaystyle\
+j_{\mathfrak{s}}(14235)k_{\mathfrak{a}}(12345)\
-j_{\mathfrak{s}}(14235)k_{\mathfrak{a}}(13245)$ $\displaystyle\
-j_{\mathfrak{s}}(14523)k_{\mathfrak{a}}(12435)\
+j_{\mathfrak{s}}(14523)k_{\mathfrak{a}}(13245)$ $\displaystyle\
-j_{\mathfrak{s}}(23145)k_{\mathfrak{a}}(13425)\
+j_{\mathfrak{s}}(24135)k_{\mathfrak{a}}(13245)\Big{)}$
$\displaystyle+a_{4}\Big{(}\ -2j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(14325)\
-2j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(12435)$ $\displaystyle\
+2j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(14325)\
+2j_{\mathfrak{s}}(23145)k_{\mathfrak{a}}(13425)$ $\displaystyle\
-3j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(12345)\
-3j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(13425)$ $\displaystyle\
+3j_{\mathfrak{s}}(13425)k_{\mathfrak{a}}(12345)\
-3j_{\mathfrak{s}}(13425)k_{\mathfrak{a}}(13245)$ $\displaystyle\
+3j_{\mathfrak{s}}(13425)k_{\mathfrak{a}}(13425)\
-3j_{\mathfrak{s}}(13425)k_{\mathfrak{a}}(14235)$ $\displaystyle\
+3j_{\mathfrak{s}}(13425)k_{\mathfrak{a}}(14325)\
+3j_{\mathfrak{s}}(13524)k_{\mathfrak{a}}(12435)$ $\displaystyle\
+3j_{\mathfrak{s}}(13524)k_{\mathfrak{a}}(13425)\
-3j_{\mathfrak{s}}(13524)k_{\mathfrak{a}}(14235)$ $\displaystyle\
-3j_{\mathfrak{s}}(13524)k_{\mathfrak{a}}(14325)\
-3j_{\mathfrak{s}}(14235)k_{\mathfrak{a}}(12345)$ $\displaystyle\
+3j_{\mathfrak{s}}(14235)k_{\mathfrak{a}}(12435)\
+3j_{\mathfrak{s}}(14235)k_{\mathfrak{a}}(13425)$ $\displaystyle\
-3j_{\mathfrak{s}}(14235)k_{\mathfrak{a}}(14235)\
-3j_{\mathfrak{s}}(14235)k_{\mathfrak{a}}(14325)$ $\displaystyle\
-3j_{\mathfrak{s}}(14523)k_{\mathfrak{a}}(12435)\
+3j_{\mathfrak{s}}(14523)k_{\mathfrak{a}}(13245)$ $\displaystyle\
+3j_{\mathfrak{s}}(23145)k_{\mathfrak{a}}(12345)\
-3j_{\mathfrak{s}}(23145)k_{\mathfrak{a}}(12435)$ $\displaystyle\
-3j_{\mathfrak{s}}(23145)k_{\mathfrak{a}}(14235)\
-3j_{\mathfrak{s}}(24135)k_{\mathfrak{a}}(12435)$ $\displaystyle\
-3j_{\mathfrak{s}}(24135)k_{\mathfrak{a}}(13425)\
+3j_{\mathfrak{s}}(24135)k_{\mathfrak{a}}(14235)$ $\displaystyle\
+3j_{\mathfrak{s}}(24135)k_{\mathfrak{a}}(14325)\
-4j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(12345)$ $\displaystyle\
+4j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(14235)\
+4j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(12345)$ $\displaystyle\
+6j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(13245)\
+6j_{\mathfrak{s}}(12534)k_{\mathfrak{a}}(13245)$ $\displaystyle\
-6j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(12435)\
-6j_{\mathfrak{s}}(14523)k_{\mathfrak{a}}(13425)$ $\displaystyle\
+6j_{\mathfrak{s}}(14523)k_{\mathfrak{a}}(14235)\
+j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(12435)$ $\displaystyle\
-j_{\mathfrak{s}}(12435)k_{\mathfrak{a}}(13425)\
-j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(13245)$ $\displaystyle\
+j_{\mathfrak{s}}(13245)k_{\mathfrak{a}}(14235)\
-j_{\mathfrak{s}}(23145)k_{\mathfrak{a}}(13245)\Big{)}$

$\begin{split}g_{1}=(12345)\quad g_{2}=(12435)\quad g_{3}=(12534)\quad\\\
g_{4}=(13245)\quad g_{5}=(13425)\quad g_{6}=(13524)\quad\\\ g_{7}=(14235)\quad
g_{8}=(14325)\quad g_{9}=(14523)\quad\\\ g_{10}=(23145)\quad
g_{11}=(24135)\quad\end{split}$ (232)

$\displaystyle\left(j_{\mathfrak{r}}\,$\mathfrak{s}$⃝\,k_{\mathfrak{s}}\right)=a_{1}\Big{(}2j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{2})\
-2j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{4})\
-2j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{2})\
+2j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{4})\
-3j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{3})$ $\displaystyle\
-3j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{10})\
+j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{1})\
+j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{4})\
+j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{1})$ $\displaystyle\
+j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{10})\
-j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{2})\
-j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{1})\
+j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{10})\
-j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{1})$ $\displaystyle\
+j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{r}}(g_{8})k_{\mathfrak{s}}(g_{1})\
+j_{\mathfrak{r}}(g_{8})k_{\mathfrak{s}}(g_{2})\
+j_{\mathfrak{r}}(g_{8})k_{\mathfrak{s}}(g_{4})\Big{)}$
$\displaystyle+a_{2}\Big{(}\ 2j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{6})\
+2j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{3})\
-2j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{6})\
+2j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{9})\
-2j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{6})$ $\displaystyle\
+2j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{9})\
-j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{4})\
+j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{7})\
-j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{9})$ $\displaystyle\
+j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{11})\
-j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{2})\
-j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{5})\
+j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{6})$ $\displaystyle\
-j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{9})\
-j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{10})\
+j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{4})\
-j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{5})\
-j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{7})$ $\displaystyle\
+j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{2})\
+j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{5})\
-j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{7})\
+j_{\mathfrak{r}}(g_{8})k_{\mathfrak{s}}(g_{5})$ $\displaystyle\
+j_{\mathfrak{r}}(g_{8})k_{\mathfrak{s}}(g_{7})\Big{)}$
$\displaystyle+a_{3}\Big{(}\ -j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{1})\
+j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{6})\
+j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{8})\
+j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{9})$ $\displaystyle\
-j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{1})\
+j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{6})\
+j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{8})\
+j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{9})$ $\displaystyle\
+j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{1})\
-j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{6})\
-j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{8})\
-j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{9})$ $\displaystyle\
+j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{1})\
-j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{6})\
-j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{8})\
-j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{9})$ $\displaystyle\
+j_{\mathfrak{r}}(g_{8})k_{\mathfrak{s}}(g_{8})\Big{)}$
$\displaystyle+a_{4}\Big{(}\ -j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{6})\
-j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{9})\
-j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{6})$ $\displaystyle\
-j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{9})\
+j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{6})\
+j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{9})\
+j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{3})$ $\displaystyle\
-j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{6})\
+j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{9})\
+j_{\mathfrak{r}}(g_{8})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{r}}(g_{8})k_{\mathfrak{s}}(g_{6})\
+j_{\mathfrak{r}}(g_{8})k_{\mathfrak{s}}(g_{9})\Big{)}$
$\displaystyle+a_{5}\Big{(}\ -2j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{2})\
-2j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{4})\
-j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{1})\
-j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{10})\
+j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{2})$ $\displaystyle\
-j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{1})\
-j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{4})\
+j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{1})\
-j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{10})$ $\displaystyle\
+j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{1})\
+j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{10})\
-j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{r}}(g_{8})k_{\mathfrak{s}}(g_{1})$ $\displaystyle\
+j_{\mathfrak{r}}(g_{8})k_{\mathfrak{s}}(g_{10})\
+j_{\mathfrak{r}}(g_{8})k_{\mathfrak{s}}(g_{3})\Big{)}$
$\displaystyle+a_{6}\Big{(}\ -2j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{9})\
-2j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{3})\
-2j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{9})\
-j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{1})\
-j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{10})$ $\displaystyle\
+j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{11})\
+j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{5})\
-j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{6})\
+j_{\mathfrak{r}}(g_{2})k_{\mathfrak{s}}(g_{9})$ $\displaystyle\
-j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{1})\
+j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{7})\
+j_{\mathfrak{r}}(g_{4})k_{\mathfrak{s}}(g_{9})\
+j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{1})\
-j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{11})$ $\displaystyle\
-j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{2})\
-j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{r}}(g_{5})k_{\mathfrak{s}}(g_{6})\
+j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{1})\
+j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{10})$ $\displaystyle\
-j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{11})\
-j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{4})\
+j_{\mathfrak{r}}(g_{7})k_{\mathfrak{s}}(g_{6})\
+j_{\mathfrak{r}}(g_{8})k_{\mathfrak{s}}(g_{1})\
+j_{\mathfrak{r}}(g_{8})k_{\mathfrak{s}}(g_{11})$ $\displaystyle\
+j_{\mathfrak{r}}(g_{8})k_{\mathfrak{s}}(g_{6})\Big{)}$

$\displaystyle\left(j_{\mathfrak{s}}\,$\mathfrak{s}$⃝\,k_{\mathfrak{s}}\right)=a_{1}\Big{(}-2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{1})\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{8})\
+2j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{1})\
+j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{6})$ $\displaystyle\
+j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{9})\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{1})\
-j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{1})\
+j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{1})\Big{)}$
$\displaystyle+a_{2}\Big{(}\ -j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{1})\
+j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{6})\
+j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{8})\
+j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{9})$ $\displaystyle\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{8})\
-j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{8})\
+j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{1})\
-j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{6})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{9})\
+j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{8})\Big{)}$
$\displaystyle+a_{3}\Big{(}\ 2j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{3})\
-2j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{6})\
+2j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{9})\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{6})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{9})\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{6})\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{9})\
-j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{3})$ $\displaystyle\
+j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{6})\
-j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{9})\
+j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{6})\
+j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{9})\Big{)}$
$\displaystyle+a_{4}\Big{(}\
-2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{11})\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{2})\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{4})\
-2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{5})\
-2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{6})$ $\displaystyle\
-2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{7})\
+2j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{10})\
-2j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{2})\
+2j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{3})\
-2j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{4})$ $\displaystyle\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{10})\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{2})\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{4})\
-j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{10})$ $\displaystyle\
+j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{2})\
-j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{4})\
+j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{10})\
-j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{2})$ $\displaystyle\
+j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{4})\Big{)}$
$\displaystyle+a_{5}\Big{(}\ -j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{11})\
+j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{2})\
+j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{4})\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{5})\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{6})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{7})\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{11})\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{2})\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{4})\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{5})$ $\displaystyle\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{6})\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{7})\
-j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{11})\
+j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{2})\
+j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{4})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{5})\
-j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{6})\
-j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{7})\
+j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{10})\
-j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{2})$ $\displaystyle\
+j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{4})\
+j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{11})\
-j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{2})\
-j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{4})$ $\displaystyle\
+j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{5})\
+j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{6})\
+j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{7})\Big{)}$
$\displaystyle+a_{6}\Big{(}\ 2j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{7})\
+2j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{8})\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{11})\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{5})\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{6})$ $\displaystyle\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{7})\
-2j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{10})\
-2j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{4})\
-2j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{8})\
-2j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{9})$ $\displaystyle\
+2j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{5})\
+2j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{8})\
-2j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{11})\
-2j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{2})\
-2j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{8})$ $\displaystyle\
-2j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{9})\
+3j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{6})\
-4j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{1})\
-j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{2})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{4})\
-j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{5})\
-j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{6})\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{10})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{2})\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{4})\
+j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{1})\
+j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{11})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{7})\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{1})\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{10})\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{11})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{2})\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{4})\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{7})\
+j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{1})\
+j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{10})$ $\displaystyle\
+j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{5})\Big{)}$
$\displaystyle+a_{7}\Big{(}\
-2j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{2})\
-2j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{7})\
-2j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{10})\
-2j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{4})\
-2j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{5})$ $\displaystyle\
+2j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{11})\
-2j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{6})\
-j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{4})\
+j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{5})$ $\displaystyle\
+j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{9})\
-j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{10})\
-j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{11})\
-j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{4})\
+j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{6})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{7})\
+j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{9})\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{11})\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{2})\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{3})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{6})\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{7})\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{9})\
-j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{10})\
-j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{2})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{5})\
+j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{9})\Big{)}$
$\displaystyle+a_{8}\Big{(}\
2j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{11})\
+2j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{2})\
+2j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{3})\
+2j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{7})\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{8})$ $\displaystyle\
-2j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{5})\
-2j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{6})\
+2j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{10})\
+2j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{4})\
+2j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{5})$ $\displaystyle\
+2j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{6})\
-2j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{11})\
-2j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{7})\
-j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{10})\
-j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{4})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{5})\
-j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{6})\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{1})\
+j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{6})$ $\displaystyle\
+j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{9})\
+j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{10})\
+j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{11})\
+j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{2})\
+j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{3})$ $\displaystyle\
+j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{4})\
+j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{7})\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{11})\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{2})\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{3})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{7})\
+j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{10})\
+j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{2})\
+j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{4})$ $\displaystyle\
+j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{5})\
+j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{6})\Big{)}$
$\displaystyle+a_{9}\Big{(}\ 2j_{\mathfrak{s}}(g_{11})k_{\mathfrak{s}}(g_{1})\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{11})\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{5})\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{6})\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{7})$ $\displaystyle\
+2j_{\mathfrak{s}}(g_{5})k_{\mathfrak{s}}(g_{1})\
+2j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{1})\
+2j_{\mathfrak{s}}(g_{7})k_{\mathfrak{s}}(g_{1})\
-3j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{2})\
-3j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{3})$ $\displaystyle\
-3j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{10})\
-3j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{4})\
-3j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{10})\
-3j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{4})\
-3j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{2})$ $\displaystyle\
-3j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{1})\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{10})\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{2})\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{3})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{4})\
-j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{1})\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{1})\
-j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{1})\Big{)}$
$\displaystyle+a_{10}\Big{(}\
-2j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{3})\
-2j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{10})\
-j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{2})\
+j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{5})\
+j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{6})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{11})k_{\mathfrak{s}}(g_{2})\
+j_{\mathfrak{s}}(g_{11})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{10})\
-j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{11})\
-j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{7})$ $\displaystyle\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{11})\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{4})\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{7})\
-j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{5})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{6})\
+j_{\mathfrak{s}}(g_{5})k_{\mathfrak{s}}(g_{10})\
-j_{\mathfrak{s}}(g_{5})k_{\mathfrak{s}}(g_{4})\
+j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{10})\
-j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{4})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{7})k_{\mathfrak{s}}(g_{2})\
+j_{\mathfrak{s}}(g_{7})k_{\mathfrak{s}}(g_{3})\Big{)}$
$\displaystyle+a_{11}\Big{(}\
2j_{\mathfrak{s}}(g_{11})k_{\mathfrak{s}}(g_{4})\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{11})\
-2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{2})\
-2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{4})\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{5})$ $\displaystyle\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{6})\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{7})\
+2j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{4})\
-2j_{\mathfrak{s}}(g_{5})k_{\mathfrak{s}}(g_{3})\
-2j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{4})$ $\displaystyle\
-2j_{\mathfrak{s}}(g_{7})k_{\mathfrak{s}}(g_{10})\
-2j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{10})\
+2j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{2})\
-2j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{3})\
+2j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{4})$ $\displaystyle\
+2j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{2})\
+2j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{4})\
-j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{4})\
-j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{5})\
-j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{6})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{10})\
+j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{11})\
+j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{3})\
-j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{4})\
+j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{7})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{11})\
+j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{2})\
-j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{7})\
+j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{10})\
-j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{2})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{5})\
+j_{\mathfrak{s}}(g_{4})k_{\mathfrak{s}}(g_{6})\Big{)}$
$\displaystyle+a_{12}\Big{(}\ 2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{3})\
-2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{6})\
+2j_{\mathfrak{s}}(g_{1})k_{\mathfrak{s}}(g_{9})\
-2j_{\mathfrak{s}}(g_{3})k_{\mathfrak{s}}(g_{5})\
-2j_{\mathfrak{s}}(g_{5})k_{\mathfrak{s}}(g_{3})$ $\displaystyle\
+2j_{\mathfrak{s}}(g_{5})k_{\mathfrak{s}}(g_{6})\
-2j_{\mathfrak{s}}(g_{5})k_{\mathfrak{s}}(g_{9})\
+2j_{\mathfrak{s}}(g_{6})k_{\mathfrak{s}}(g_{5})\
-2j_{\mathfrak{s}}(g_{7})k_{\mathfrak{s}}(g_{9})\
-2j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{5})$ $\displaystyle\
-2j_{\mathfrak{s}}(g_{9})k_{\mathfrak{s}}(g_{7})\
-4j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{3})\
+4j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{6})\
-4j_{\mathfrak{s}}(g_{8})k_{\mathfrak{s}}(g_{9})\
+j_{\mathfrak{s}}(g_{10})k_{\mathfrak{s}}(g_{5})$ $\displaystyle\
-j_{\mathfrak{s}}(g_{11})k_{\mathfrak{s}}(g_{5})\
+j_{\mathfrak{s}}(g_{11})k_{\mathfrak{s}}(g_{6})\
+j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{3})\
+j_{\mathfrak{s}}(g_{2})k_{\mathfrak{s}}(g_{4})\