# Infinitesimal deformations of symmetric simple modular Lie algebras and Lie
superalgebras

Sofiane Bouarroudj1, Pavel Grozman2, Dimitry Leites3 1Department of
Mathematics, United Arab Emirates University, Al Ain, PO. Box: 17551;
<EMAIL_ADDRESS>
2Equa Simulation AB, Stockholm, Sweden<EMAIL_ADDRESS>
3A.Salam International School of Mathematical Science Lahore, Pakistan,
on leave from Department of Mathematics, University of Stockholm, Roslagsv.
101, Kräftriket hus 6, SE-106 91 Stockholm, Sweden<EMAIL_ADDRESS>
###### Abstract.

Over algebraically closed fields of positive characteristic, infinitesimal
deformations of simple finite dimensional symmetric (the ones that with every
root have its opposite of the same multiplicity) Lie algebras and Lie
superalgebras are described for small ranks. The results are obtained by means
of the Mathematica based code SuperLie.

The infinitesimal deformation given by any odd cocycle is integrable. The
moduli of the deformations form, in general, a supervariety. Not each even
cocycle is integrable; but for those that are integrable, the global deforms
(the results of deformations) are linear with respect to the parameter.

In characteristic 2, the simple 3-dimensional Lie algebra admits a parametric
family of non-isomorphic simple deforms.

Some of Shen’s ”variations of G(2) theme” are interpreted as two global
deforms corresponding to the several of the 20 infinitesimal deforms first
found by Chebochko; we give their explicit form.

###### Key words and phrases:

modular Lie superalgebra

###### 1991 Mathematics Subject Classification:

17B50, 70F25

We are thankful to A. Lebedev for help; DL is thankful to MPIMiS, Leipzig, for
financial support and most creative environment. We are thankful to N.
Chebochko and M. Kuznetsov for helpful discussions of their unpublished
results pertaining to this paper.

##  1\. Introduction

Hereafter, ${\mathbb{K}}$ is an algebraically closed field of characteristic
$p>0$ and ${\mathfrak{g}}$ is a finite dimensional Lie (super)algebra;
${\mathbb{Z}}_{+}$ is the set of non-negative integers.

###  1.1. The quest for simple modular Lie algebras. The Kostrikin–Shafarevich
conjecture

Shafarevich, together with his student Kostrikin, first considered the
restricted simple modular Lie algebras. This self-restriction was, perhaps,
occasioned by the fact that only restricted Lie algebras correspond to
algebraic groups, and being a geometer, albeit an algebraic one, Shafarevich
did not see much reason to consider non-restricted Lie algebras. Recently, on
a different occasion, Deligne gave us an advice [LL] which we interpret as a
suggestion to look, if $p>0$, at the groups (geometry) rather than at Lie
algebras. Since only restricted Lie algebras correspond to algebraic groups,
we interpret this advice as a certain natural restriction of the
classification problem of simple Lie (super)algebras which makes the problem
more tangible but requires to select deforms (the results of deformations)
with $p$-structure as precious gems.

Be as it may, in late 1960s, Kostrikin and Shafarevich formulated a conjecture
describing all simple finite dimensional (not only restricted) modular Lie
algebras over algebraically closed fields of characteristic $p>7$. The
statement of the conjecture turned out to hold for $p=7$ as well.

After 30 years of work of several teams of researchers, Block, Wilson, Premet
and Strade not only proved the KSh-conjecture, but even completed the
classification of all simple finite dimensional Lie algebra over algebraically
closed fields of characteristic $p>3$, see [S].

It turns out that, for $p>3$, the classification can be formulated as follows
(for details and definitions, see [BGL, Lt]):

(1) Take any finite dimensional Lie algebra of the form ${\mathfrak{g}}(A)$
with indecomposable Cartan matrix $A$, or a trivial central extension
${\mathfrak{c}}({\mathfrak{g}}(A))$ thereof. In one of the
${\mathbb{Z}}$-gradings of ${\mathfrak{g}}:={\mathfrak{g}}(A)$ or
${\mathfrak{c}}({\mathfrak{g}}(A))$, take the Cartan-Tanaka-Shchepochkina
(CTS) prolong $({\mathfrak{g}}_{-},{\mathfrak{g}}_{0})_{\ast,\underline{N}}$
of the pair $({\mathfrak{g}}_{-},{\mathfrak{g}}_{0})$ whose dimension is
bounded by a shearing parameter $\underline{N}$. All simple finite dimensional
modular Lie algebras are among such prolongs or the quotients of their derived
algebras modulo the center, and deformations thereof.

Although for $p=2,3$, and for Lie superalgebras, certain other types of pairs
$({\mathfrak{g}}_{-},{\mathfrak{g}}_{0})$ also yield simple examples, the
importance of the algebras of the form ${\mathfrak{g}}(A)$ is clear; it is
also of independent interest.

Weisfeiler and Kac [WK] gave a classification of finite dimensional Lie
algebras ${\mathfrak{g}}(A)$ with indecomposable Cartan matrix $A$ for any
$p>0$, but although the idea of their proof is OK, the paper has several gaps
and vague notions (the Brown algebra ${\mathfrak{br}}(3)$ was missed [Br3,
KWK]; the definition of the Lie algebra with Cartan matrix given in [K] (and
applicable to Lie superalgebras and modular Lie (super)algebras) was not
properly developed at the time [WK] was written; the algebras
${\mathfrak{g}}(A)^{(1)}/{\mathfrak{c}}$ — which have no Cartan matrix — are
sometimes identified as having one, and so on). All these notions, and several
more, are clarified in [BGL, LCh].

There were also known several scattered examples of serial and (conjecturally,
in the absence of a classification) exceptional simple modular Lie algebras
for $p=3$ and 2, see [Lt]. For all these cases, there remained the problem of
description of deformations. In the latest paper on the topic, with difficult
results nicely explained, Natasha Chebochko gave an overview of the situation
for Lie algebras of the form ${\mathfrak{g}}(A)$ and
${\mathfrak{g}}(A)^{(1)}/{\mathfrak{c}}$. She writes:

(2) “According to [Dz], for $p=3$, the Lie algebra $C_{2}$ is the only algebra
among the series $A_{n}$, $B_{n}$, $C_{n}$, $D_{n}$ that admits non-trivial
deformations. In [Ru] it was established that over a field of characteristic
$p>3$ all the classical Lie superalgebras are rigid. In [KuCh] and [KKCh] a
new scheme was proposed for studying rigidity, and it was proved that the
classical Lie algebras of all types over a field of characteristic $p>2$ are
rigid, except for the Lie algebra of type $C_{2}$ for $p=3$. (“Classical” here
means any algebra with indecomposable Cartan matrix (or its quotient modulo
center), except Brown algebras: for ${\mathfrak{br}}(2)$ deformations were
known, for ${\mathfrak{br}}(3)$ the question was open. This makes our answer
concerning the Brown algebras ${\mathfrak{br}}(3)$ even more interesting.
BGL.) For $p=2$, some deformations of the Lie algebra of type $G_{2}$ were
constructed in [She1]. … The Lie algebras of type $A_{l}$ for $l+1\equiv
0\pmod{2}$, $D_{l}$ and $E_{7}$, have non-trivial centers. We shall say that
the corresponding quotient algebras are of type $\overline{A_{l}}$,
$\overline{D_{l}}$ and $\overline{E_{7}}$, respectively. …

(3) Theorem. Let $L$ be a Lie algebra over a field of characteristic $2$.
$(1)$ If $L$ is one of the types $A_{l}$ for $l>1$, $D_{l}$ for $l\equiv
1\pmod{2}$, $E_{6}$, $E_{7}$, $E_{8}$, $\overline{A_{l}}$ for $l\neq 3,5$, or
$\overline{D_{l}}$ for $l\equiv 0\pmod{2}$ and $l\neq 6$, then $H^{2}(L;L)=0$.
$(2)$ If $L$ is of type $\overline{A_{l}}$ for $l=3,5$, then $\dim
H^{2}(L;L)=20$.
$(3)$ If $L$ is of type $D_{4}$, then $\dim H^{2}(L;L)=24$.
$(4)$ If $L$ is of type $D_{l}$ for $l\equiv 0\pmod{2}$ and $l>4$ or
$\overline{D_{l}}$ for $l\equiv 1\pmod{2}$, then $\dim H^{2}(L;L)=2l$.
$(5)$ If $L$ is of type $\overline{D_{6}}$, then $\dim H^{2}(L;L)=64$.
$(6)$ If $L$ is of type $\overline{E_{7}}$, then $\dim H^{2}(L;L)=56$.”

####  1.1.1. Elucidating certain moments unclear to us in the conventional
presentations

1) The term “classical” in the above quotation (and almost all other papers
and books on the topic) is applied to simple Lie algebras with Cartan matrix
of the same types that exist for $p=0$; their quotients modulo center are also
“classical”, but, for $p=2$, the simple Lie algebras of Brown, Weisfeiler and
Kac, and ${\mathfrak{o}}(2n+1)$ are left out. Clearly, the nomenclature should
be improved; Shen [She2] indicates other reasons for being unhappy with it.

Neither the $p=2$ analogs of ${\mathfrak{o}}_{I}(2n)$ nor its simple
subquotient have Cartan matrix. In the above quotation, they are not qualified
as “classical”, whereas we are sure they should be. So we have to consider

(4) ${\mathfrak{o}}_{I}(2n),\;\;{\mathfrak{o}}(2n+1)\;\text{(a.k.a. $B_{l}$),
the Brown algebras, and the Weisfeiler-Kac algebras.}$

2) The simple Lie algebras constitute a natural “core”, but some of their
relatives are “better” (are restricted, possess Cartan matrix, and are rigid,
whereas their simple cores may lack one or all of these features). In
particular, the abundance of deformations might point at an inner “flaw” of
the object whose relative’s behavior is impeccable …

Together with Lebedev, in [BGL] and in chapters in [LCh] due to Lebedev, we
clarified the above moments (we will reproduce the needed definitions in what
follows).

####  1.1.2. Explicit cocycles v. ecology

Having written the first 15 pages of this paper we were appalled by the amount
of the space saturated by explicit cocycles. Who needs them?! Let us give just
the dimensions and save paper! However, it was only thanks to the explicit
form of the cocycles that we were able to interpret some of the mysterious
Shen’s “variations”. And only having the explicit form of the cocycle can one
check if the local deformation is integrable or be able to compute the global
one.

###  1.2. Main result

In [BGL], we have classified finite dimensional modular Lie superalgebras of
the form ${\mathfrak{g}}(A)$ with indecomposable symmetrizable Cartan matrix
$A$ over algebraically closed fields, in particular, the simple Lie
superalgebras of the form ${\mathfrak{g}}(A)$ and those for which
${\mathfrak{g}}(A)^{(1)}/{\mathfrak{c}}$, where ${\mathfrak{c}}$ is the
center, is simple. For brevity, we will sometimes address the latter algebras
also as having Cartan matrix, though, strictly speaking, they do not possess
any Cartan matrix, see Warning in [BGL]. In [Ch], Chebochko described
infinitesimal deformations of the “classical” Lie algebras with Cartan matrix,
except for (4) for which it remained an open problem.

The Lie superalgebras with Cartan matrix, and the queer algebras
${\mathfrak{psq}}(n)$ for $p\neq 2$ are “symmetric”:

(5) With each root $\alpha$ they have a root $-\alpha$ of the same
multiplicity.

(If $p=2$, there are more examples of queer type Lie superalgebras, serial and
exceptional, and some of them are not symmetric, see [LCh].)

In this note, a sequel to [BGL], we describe deformations of “symmetric” Lie
(super)algebras:

(1) the “left out” cases (4), and of Lie superalgebras of small rank, both
with Cartan matrix (and relatives thereof) classified in [BGL],

(2) of the so-called queer type,

(3) of certain Lie algebras without Cartan matrix: ${\mathfrak{o}}_{I}(2n)$,
leaving its super versions without Cartan matrix
(${\mathfrak{o}}{\mathfrak{o}}_{II}(2n+1|2m+1)$,
${\mathfrak{o}}{\mathfrak{o}}_{II}(2n|2m)$,
${\mathfrak{o}}{\mathfrak{o}}_{I\Pi}(2n|2m)$,
${\mathfrak{o}}{\mathfrak{o}}_{II}(2n+1|2m)$) for a time being; we intend to
do this elsewhere.

Conjecturally, the symmetric algebras of higher ranks over fields of
characteristic distinct from 2 are rigid, except for the simple relatives of
Lie (super)algebras of Hamiltonian vector fields; for the computers available
to us at the moment, the problem is out of reach even with this code.

####  1.2.1. On Shen’s “variations”

Trying to append the results of Chebochko [Ch] by deforms of the Lie algebras
she did not consider but which we consider no less “classical” than the ones
she considered, we obtained, as a byproduct, an elucidation of Shen’s
“variations” [She1, LLg]. Shen described seven “variations” $V_{i}G(2)$ of
${\mathfrak{g}}(2)$ and three more “variations” of ${\mathfrak{sl}}(3)$; Shen
claimed that all his examples are simple and all but two
($V_{1}G(2):={\mathfrak{g}}(2)$ and $V_{7}G(2):={\mathfrak{wk}}(3;a)$) are
new. It was later found that the “variations” of ${\mathfrak{sl}}(3)$ are
isomorphic to ${\mathfrak{sl}}(3)$, the variations of dimension 15 are not
simple (this has to be verified). At the moment, the only verified claim on
simple and really new Lie algebras Shen has discovered concerns the
2-parameter deformations of ${\mathfrak{g}}(2)\simeq{\mathfrak{psl}}(4)$.
Several “variations” are depicted with typos: as stated, these algebras do not
satisfy Jacobi identity or have ideals. Unfortunately, we did not guess how to
amend the multiplication tables whereas our letters to Shen or his students
remain unanswered.

Chebochko [Ch] has found all infinitesimal deformations of
${\mathfrak{g}}(2)\simeq{\mathfrak{psl}}(4)$ and we have found that each
individual cocycle is integrable. In [Ch1], Chebochko found out that the space
of cocycles constitutes two (apart from the origin) orbits relative
$\mathop{\mathrm{{Aut\/}}}\nolimits({\mathfrak{psl}}(4))$, and described the
two respective Lie algebras.

We only consider the Lie algebras of small rank because our Mathematica-based
code SuperLie (see [Gr]) with which we got our results is incapable to process
higher ranks on computers available to us. However, since the higher the rank
the more rigid simple Lie (super)algebras are, we make the following

######  Conjecture.

In this paper we have found all (except for the analogs of the cases of
Chebochko’s theorem (3) that we have to add) the infinitesimal deformations of
finite dimensional Lie algebras and Lie superalgebras with indecomposable
Cartan matrices for $p=5$ and $3$ (and their simple subquotients) as well as
of queer Lie superalgebras (and their simple subquotients).

###  1.3. Motivations

The classification of simple finite dimensional Lie algebras ${\mathfrak{g}}$
over fields ${\mathbb{K}}$ of characteristic $p>3$ — a proof of the
generalized Kostrikin-Shafarevich conjecture — is now completed, see [S]. So
it is time now to consider the cases $p=3$, 2, and the super case, especially
in view of the hidden supersymmetry of the non-super $p=2$ case, see [ILL]).
Let us list the results known earlier.

Several examples of simple Lie algebras over fields ${\mathbb{K}}$ for $p=3$
not embraced by the approach of the Kostrikin-Shafarevich conjecture are given
in [S]; in [GL4], some of these examples, earlier considered “mysterious”, are
described as (generalized) Cartan prolongs provided they are
${\mathbb{Z}}$-graded; other examples from [S] are deformations of these
${\mathbb{Z}}$-graded ones. In [WK], the simple modular Lie algebras
possessing Cartan matrix are classified, some of them form parametric
families, but no complete study of deformations was performed.

It was clear since long ago that the smaller characteristic, the less rigid
the simple Lie algebras are; Rudakov gave an example of a 3-parameter family
of simple Lie algebras (for the first published description, see [Kos]; for an
exposition of the initial (never published) Rudakov’s approach — in terms of
the Cartan prolongation, — see [GL4]).

After Rudakov’s example became known, Kostrikin and Dzhumadildaev ([DK, Dz1,
Dz2, Dz3]) studied various (e.g., filtered and infinitesimal) deformations of
simple vectorial Lie algebras (sometimes dubbed algebras of Cartan type); for
a detailed summary of the part of their results with understandable proofs,
and some new results (all pertaining to the infinitesimal deformations), see
[Vi].

Rudakov’s paper [Ru] clearly showed that speaking about deformations it is
unnatural to consider the modular Lie algebras naively, as vector spaces: Lie
algebras should be viewed as algebras in the category of varieties. This
approach should, actually, be applied even over fields of characteristic 0,
but the simplicity of the situation with finite dimensional Lie algebras
obscures this (rather obvious) fact. Lie superalgebras can not be treated
otherwise in various problems of interest in applications, see [LSh].

Recently the deformations of the simple modular Lie algebras with Cartan
matrix, except for the Brown algebras we consider, were classified ([Ch, KK,
KuCh, KKCh]). The findings of Chebochko did not, however, elucidate several of
mysterious Shen’s “variations”, see [She1, LLg].

###  1.4. Disclaimer

1) Same as practically everybody ([Ch, KK, KuCh, KKCh, Vi]) we disregard, for
the moment, the danger of ignoring singular global deformations $=$ an
interesting but dangerous possibility pointed at by D. B. Fuchs with co-
authors [FF, FL]. This danger, however, may only concern even deformations
whereas most of the deformations we list here are odd.

2)In [Ch, KK, KuCh, KKCh], a complete description of deformations of the
“classical” simple Lie algebras with Cartan matrix is performed over fields
${\mathbb{K}}$ for $p=3$ and 2. Although in [LL] there are given reasons111The
overwhelming abundance of cocycles found in [Ch, Vi], is hardly a reason as we
thought before we have read [Ch, Ch1]: Although there are many nontrivial
cocycles, non-isomorphic deforms of ${\mathfrak{g}}$ correspond only to the
distinct orbits in the space of cocycles under the action of the Chevalley
group $\mathop{\mathrm{{Aut\/}}}\nolimits({\mathfrak{g}})$ and there are very
few such orbits. to doubt the possibility to liberally apply the conventional,
so far, definition of Lie algebra (co)homology ([Fu]) to the case where $p>0$,
these doubts seem to be inapplicable for $H^{2}({\mathfrak{g}};M)$ if
${\mathfrak{g}}$ possesses Cartan matrix, $\dim M<\infty$, and $p\neq 2$. In
particular, it seems safe to compute $H^{2}({\mathfrak{g}};{\mathfrak{g}})$
and interpret the result in terms of infinitesimal deformations.

Let us explain how definition of $U({\mathfrak{g}})$ affects the definition of
cohomology and the notion of (co)induced representations. The scientific
definition of the Lie (super)algebra cohomology is

(6)
$H^{i}({\mathfrak{g}};M):=\mathop{\mathrm{Ext}}\nolimits^{i}_{U({\mathfrak{g}})}({\mathbb{K}};M).$

So it is clear, actually, how to approach the problem, at least for the
modular Lie algebras obtained by means of the Kostrikin-Shafarevich approach
(and its super analog): Speaking about non-super cases, take any book (e.g.,
[St]) in which a convenient ${\mathbb{Z}}$-form
$U_{\mathbb{Z}}({\mathfrak{g}})$ of $U({\mathfrak{g}})$ is described for any
simple complex ${\mathfrak{g}}$, and introduce $\underline{N}$ (similar to the
$\underline{N}$ in the definition of the algebra of divided powers ${\cal
O}(n;\underline{N})$) by setting something like

(7) $\begin{array}[]{ll}U({\mathfrak{g}};\underline{N}):=&\text{subalgebra of
$U_{\mathbb{Z}}({\mathfrak{g}})$ constructed}\\\ &\text{``similarly to the
algebra of divided powers" ${\cal O}(n;\underline{N})$};\\\
H^{i}_{\underline{N}}({\mathfrak{g}};M):=&\mathop{\mathrm{Ext}}\nolimits^{i}_{U({\mathfrak{g}};\underline{N})}({\mathbb{K}};M).\end{array}$

How to perform this “similar construction” of “something like” is the whole
point.

Absolutely correct — in terms of the conventional definition (6) —
computations of Dzhumadildaev elucidated in [Vi] imply that
${\mathfrak{vect}}(n;\underline{N}):={\mathfrak{der}}({\cal
O}(n;\underline{N}))$, where only “special derivatives” are considered, is not
rigid; more precisely, ${\mathfrak{vect}}(n;\underline{N})$ has infinitesimal
deformations. We find this result “ideologically wrong” and believe that the
cause is buried in the definitions used. Recall our arguments in favor of
rigidity of ${\mathfrak{vect}}$, see [LL]:

Let ${\mathfrak{h}}$ be a subalgebra of ${\mathfrak{g}}$. For any
${\mathfrak{h}}$-module $V$, we define a series of coinduced
${\mathfrak{g}}$-modules:

(8)
$\begin{array}[]{l}\mathop{\mathrm{Coind}}\nolimits_{\mathfrak{h}}^{\mathfrak{g}}(V;\underline{N}):=\mathop{\mathrm{Hom}}\nolimits_{U({\mathfrak{h}};\underline{N})}(U({\mathfrak{g}};\underline{N}),V);\end{array}$

Then, in terms of the conjectural definition (7), we should have the following
analog of the well-known isomorphism:

(9)
$\begin{array}[]{l}H^{i}_{\underline{N}}({\mathfrak{g}};\mathop{\mathrm{Coind}}\nolimits_{\mathfrak{h}}^{\mathfrak{g}}(V;\underline{N}))\simeq
H^{i}({\mathfrak{h}};V).\end{array}$

This isomorphism would imply that, for
${\mathfrak{g}}={\mathfrak{vect}}(n;\underline{N})$, we should have

$H^{2}_{\underline{N}}({\mathfrak{g}};{\mathfrak{g}})\simeq
H^{2}({\mathfrak{gl}}(V);V)=0,\text{~{}~{}where $\dim V=n$,}$

at least, if $n$ is not divisible by $p$.

The situation is opposite, in a sense, to that with the Kac-Moody groups that
“did not exist” until a correct definition of cohomology was used; or with
Dirac’s $\delta$-function which is not a function in the conventional sense.

Dzhumadildaev [Dz2] (also Farnsteiner and Strade [FS]) showed that for $p>0$
the conventional analog of the statement (9), a.k.a. Shapiro’s lemma, should
be formulated differently because
$H^{i}({\mathfrak{g}};\mathop{\mathrm{Coind}}\nolimits_{\mathfrak{h}}^{\mathfrak{g}}(V))$
strictly contains $H^{i}({\mathfrak{h}};V)$. We hope that one can get rid of
these extra cocycles in an appropriate theory.

However, even if we are wrong here and any reasonable theory contains these
infinitesimal deformations, and nobody, as far as we know, investigated their
number after factorization by the action of the corresponding automorphism
group.

3) The Lie (super)algebra (co)homology can also be defined “naively”, as
generalizations (and dualizations) of the de Rham complex. In this approach
the enveloping algebra does not appear explicitly and the divided powers we
tried to introduce in item 2) of Disclaimer seem to disappear.

The divided powers of (co)chains naturally appear in the study of Lie
superalgebras for any $p$, even for $p=0$, but their meaning is unclear at the
moment. We suggest to denote the corresponding spaces of homology by
$DPH_{(n)}^{\underline{N}}({\mathfrak{g}};M)$ and cohomology by
$DPH^{(n)}_{\underline{N}}({\mathfrak{g}};M)$.

For $p=2$ and Lie superalgebras, there is an interpretation: Such divided
(co)chains (for $\underline{N}_{i}>1$ for every odd basis element
$X_{i}\in{\mathfrak{g}}$) are indispensable in the study of relations and
deformations of the Lie superalgebras because the bracket is now determined by
the squaring.

We denote the product of divided powers of cochains by wedge, the usual powers
are denoted $(dx)^{\wedge n}$ whereas the divided powers are denoted
$(dx)^{(\wedge n)}$. Actually, we should not write
$dx\in\Pi({\mathfrak{g}}^{*})$ but rather $x^{*}$ either for brevity or having
in mind the element of ${\mathfrak{g}}^{*}$ unless the wedge product is
needed.

4) On integrability. We compute infinitesimal deformations, i.e., we compute
$H^{2}({\mathfrak{g}};{\mathfrak{g}})$. However, if the cocycle is odd, it
certainly extends to a global deformation, albeit with an odd parameter.

Kostrikin and Dzhumadildaev ([DK]) claimed that every local (meaning
infinitesimal) deformation of the Lie algebra $W_{1}(m)$ (we denote this
algebra by ${\mathfrak{vect}}(1;\underline{m})$) is integrable.222A question
arises: in which sense is the classification of [S] complete?! It turns out:
“as stated”: for $p>3$, ALL simple finite dimensional algebras are classified
because although there are not only filtered deformations of Lie algebras of
vectorial type, the deforms are isomorphic to something known. Ain’t it a
miracle: if $p=2$, this definitely is not so. We were unable to follow their
proof (reduction to a paper of R. Amayo [Proc. London Math. Soc. (3) 33
(1976), no. 1, 28–64; MR0409573 (53 #13327a) ,b]) but, to our incredulity,
have encountered a similar phenomenon with most of the cocycles unearthed in
this paper: Most of the infinitesimal deformations of some of the Lie
(super)algebras considered here are integrable; moreover, the global
deformation corresponding to a given (homogeneous with respect to weight)
cocycle $c$ is often linear in parameter, i.e., it is of the form

(10) $[x,y]_{\text{new}}=[x,y]+\alpha c(x,y).$

The deformation of ${\mathfrak{osp}}(4|2)$ and Shen’s deformations of
${\mathfrak{psl}}(4)$ are linear in parameter, and so the corresponding
cocycles give the global deformations. These facts encouraged us to conjecture
that ALL cocycles we have found are integrable (this is not so) and directly
investigate which of the other cocycles determine the new bracket (10).

We also have to consider deformations of the deforms (we intend to do this
elsewhere).

####  1.4.1. What ${\mathfrak{g}}(A)$ is

The Lie (super)algebras of the form ${\mathfrak{g}}(A)$ (sometimes called,
together with (super)algebras of certain other types, “contragredient”) are
determined as follows. Let $A=(A_{ij})$ be an arbitrary $n\times n$ matrix of
rank $l$ with entries in ${\mathbb{K}}$. Fix a vector space ${\mathfrak{h}}$
of dimension $2n-l$ and its dual ${\mathfrak{h}}^{*}$, select $n$ linearly
independent vectors $h_{i}\in{\mathfrak{h}}$ and $n$ linearly independent
vectors $\alpha_{j}\in{\mathfrak{h}}^{*}$ so that $\alpha_{i}(h_{j})=A_{ij}$.

Let $I=\\{i_{1},\dots,i_{n}\\}\subset({\mathbb{Z}}/2{\mathbb{Z}})^{n}$;
consider the free Lie superalgebra $\widetilde{{\mathfrak{g}}}(A,I)$ generated
by $e_{1}^{\pm},\dots,e_{n}^{\pm}$, where $p(e_{j}^{\pm})=i_{j}$, and
${\mathfrak{h}}$, subject to relations (here either all superscripts $\pm$ are
$+$ or all are $-$)

(11)
$[e_{i}^{+},e_{j}^{-}]=\delta_{ij}h_{i};\quad[h,e_{j}^{\pm}]=\pm\alpha_{j}(h)e_{j}^{\pm}\text{~{}~{}for
any $h\in{\mathfrak{h}}$ and any
$i,j$};\quad\;[{\mathfrak{h}},{\mathfrak{h}}]=0.$

Let $\tilde{\mathfrak{g}}_{\pm}$ be subalgebras generated by $e_{i}^{\pm}$.

######  Statement.

There exists a maximal ideal ${\mathfrak{r}}$ among the ideals of
$\tilde{\mathfrak{g}}(A,I)$ whose intersection with ${\mathfrak{h}}$ is zero,
and ${\mathfrak{r}}$ is the direct sum of the ideals
${\mathfrak{r}}\bigcap\tilde{\mathfrak{g}}_{+}$ and
${\mathfrak{r}}\bigcap\tilde{\mathfrak{g}}_{-}$.

The statement is well known over ${\mathbb{C}}$ for the finite dimensional and
certain infinite dimensional simple Lie algebras [K]; for an explicit
description of the ideal ${\mathfrak{r}}$ for simple Lie superalgebras over
${\mathbb{C}}$ of the same type, see [GL1]; for the modular case, see
Lebedev’s chapters in [LCh].

The change

$A\longmapsto\tilde{A}:=\mathop{\mathrm{diag}}\nolimits(\lambda_{1},\dots,\lambda_{n})A,\text{~{}~{}where
$\lambda_{1},\dots,\lambda_{n}\in{\mathbb{K}}\setminus\\{0\\}$},$

preserves the Lie (super)algebra $\tilde{\mathfrak{g}}(A,I)$. The Cartan
matrix $A$ is said to be normalized if $A_{jj}=0$ or 1, or 2 (only if
$i_{j}={\bar{0}}$, where ${\bar{0}}$ and ${\bar{1}}$ are residue classes
modulo 2); in order to distinguish the cases $i_{j}={\bar{0}}$ from
$i_{j}={\bar{1}}$, we write $A_{jj}={\bar{0}}$ or ${\bar{1}}$, instead of 0 or
1, if $i_{j}={\bar{0}}$. We will only consider normalized Cartan matrices; for
them, we can skip $I$.

The subalgebra ${\mathfrak{h}}$ diagonally acts on $\tilde{\mathfrak{g}}(A)$
and the nonzero eigenspaces with nonzero eigenvalues are called roots. In the
modular case, it is more appropriate to use the following definition. The
algebra $\tilde{\mathfrak{g}}(A)$, where $A$ is of size $n\times n$, is
naturally ${\mathbb{Z}}^{n}$-graded, where $\deg e_{i}^{\pm}=(0,\dots,0,\pm
1,0,\dots,0)$, where $\pm 1$ occupies the $i$th slot. The nonzero eigenspaces
with nonzero eigenvalues in this grading are said to be roots. (Over
${\mathbb{C}}$, this grading is equivalent to the usual grading by weights.)
For any subset $B=\\{\sigma_{1},\dots,\sigma_{n}\\}\subset R$, we set:

$R_{B}^{\pm}=\\{\alpha\in R\mid\alpha=\pm\sum
n_{i}\sigma_{i},\;\;n_{i}\in{\mathbb{Z}}_{+}=\\{0,1,2,\dots\\}\\}.$

The subset $B\subset R_{B}^{+}$ is said to be a system of simple roots of $R$
(or ${\mathfrak{g}}$) if $\sigma_{1},\dots,\sigma_{n}$ are linearly
independent and $R=R_{B}^{-}\coprod R_{B}^{+}$.

Set ${\mathfrak{g}}(A):=\tilde{\mathfrak{g}}(A)/{\mathfrak{r}}$. Observe that
the conditions sufficient for simplicity of Lie algebras ${\mathfrak{g}}(A)$
over ${\mathbb{C}}$ fail to ensure simplicity of modular Lie algebras
${\mathfrak{g}}(A)$ and Lie superalgebras ${\mathfrak{g}}(A)$.

######  Warning.

For $m-n\equiv 0\pmod{p}$ (for $m=n$ over ${\mathbb{C}}$), neither
${\mathfrak{sl}}(m|n)$ nor its simple quotient333If a Lie superalgebra
${\mathfrak{g}}\subset{\mathfrak{gl}}(V)$ contains the space of scalar
matrices ${\mathfrak{s}}={\mathbb{K}}1_{m|n}$, we denote
${\mathfrak{p}}{\mathfrak{g}}={\mathfrak{g}}/{\mathfrak{s}}$ the projective
version of ${\mathfrak{g}}$. ${\mathfrak{psl}}(m|n)$ possess Cartan matrix; it
is ${\mathfrak{gl}}(m|n)$ which possesses it.

##  2\. The queer series

In the text-book [Ls] it is demonstrated that there are two superizations of
${\mathfrak{gl}}(n)$: a naive one, ${\mathfrak{gl}}(n|m)$, and the “queer”
one, ${\mathfrak{q}}(n)$ which preserves an “odd” complex structure. Recall
that

${\mathfrak{q}}(n):=\left\\{X\in{\mathfrak{gl}}(n|n)\mid[X,J_{2n}]=0,\;\text{where
$J_{2n}=\begin{pmatrix}0&1_{n}\\\ -1_{n}&0\end{pmatrix}$}\right\\};$

explicitly,

${\mathfrak{q}}(n):=\left\\{\begin{pmatrix}A&B\\\ B&A\end{pmatrix}\right\\}.$

We set

${\mathfrak{sq}}(n):=\left\\{\begin{pmatrix}A&B\\\ B&A\end{pmatrix}\text{ such
that $\mathrm{tr}(B)=0$}\right\\}.$

Let $\mathfrak{psq}(n):=\mathfrak{sq}(n)/\mathfrak{center}$ be the
projectivisation of $\mathfrak{sq}(n)$. We denote the images of the
$A_{ij}$-elements in $\mathfrak{psq}(n)$ by $a_{ij}$ and the images of the
$B_{ij}$-elements by $b_{ij}$.

It is not difficult to show that the derived superalgebra
$\mathfrak{psq}^{(1)}(n)=[\mathfrak{psq}(n),\mathfrak{psq}(n)]$ is simple and
is only different from $\mathfrak{psq}(n)$ for $n=pk$, when $\mathfrak{sq}(n)$
contains $\begin{pmatrix}0&1_{n}\\\ -1_{n}&0\end{pmatrix}$.

##  3\. Results

Observe that, for our problem, it does not matter which of several Cartan
matrices that a given algebra ${\mathfrak{g}}(A)$ possesses we take: The
simplest incarnation will do.

Since our algebras are symmetric with respect to the change of the sign of the
roots, it suffices to consider cocycles of only non-negative (or non-positive)
degrees. We do not list cocycles of positive (negative) degrees.

We denote the positive Chevalley generators by $x_{i}$, the corresponding
negative ones by the $y_{i}$.

We underline the degrees of the odd cocycles.

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{br}}(2;\alpha)$, where $\alpha\neq 0,-1$, the
Cartan matrix is

(12) $\begin{pmatrix}2&-1\\\ \alpha&2\end{pmatrix}\quad\text{and the basis
}\;x_{1},\;x_{2},\;x_{3}=[x_{1},x_{2}],\quad
x_{4}=-\mathop{\mathrm{ad}}\nolimits_{x_{2}}^{2}(x_{1}).$

Then $H^{2}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycles
$(\ref{eqbr2a})$.

(13) $\tiny\begin{array}[]{ll}\deg=-6:&2\alpha(1+\alpha)y_{1}\otimes
dx_{3}{\wedge}dx_{4}+(1+\alpha)y_{3}\otimes dx_{1}{\wedge}dx_{4}+y_{4}\otimes
dx_{1}{\wedge}dx_{3}\\\ &\\\ \deg=-3:&\alpha(1+\alpha)x_{1}\otimes
dx_{2}{\wedge}dx_{4}+2(1+\alpha)y_{2}\otimes dx_{4}{\wedge}dy_{1}+y_{4}\otimes
dx_{2}{\wedge}dy_{1}\\\ &\\\ \deg=0:&h_{1}\otimes dx_{2}\wedge
dy_{2}+h_{1}\otimes dx_{3}\wedge dy_{3}+x_{1}\otimes dx_{3}\wedge
dy_{2}-y_{1}\otimes dx_{2}\wedge dy_{3}\end{array}$

######  Lemma.

For
${\mathfrak{g}}={\mathfrak{br}}(2)=\mathop{\lim}\limits_{-\frac{2}{a}\longrightarrow
0}~{}{\mathfrak{br}}(2,a)$, the Cartan matrix is

(14) $\begin{pmatrix}2&-1\\\ -1&{\bar{0}}\end{pmatrix}\quad\text{and the basis
}\;x_{1},\;x_{2},\;x_{3}=[x_{1},x_{2}],\quad
x_{4}=-\mathop{\mathrm{ad}}\nolimits_{x_{2}}^{2}(x_{1}).$

Then $H^{2}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycles
$(\ref{eqbr2})$.

(15) $\tiny\begin{array}[]{ll}\deg=-6:&y_{1}\otimes
dx_{3}{\wedge}dx_{4}+y_{3}\otimes dx_{1}{\wedge}dx_{4}+2\,y_{4}\otimes
dx_{1}{\wedge}dx_{3};\\\ \deg=-3:&{x_{1}\otimes
dx_{2}{\wedge}dx_{4}+y_{2}\otimes dx_{4}{\wedge}dy_{1}+y_{4}\otimes
dx_{2}{\wedge}dy_{1}};\\\ \deg=0:&h_{1}\otimes
dx_{2}{\wedge}dy_{2}+h_{1}\otimes dx_{3}{\wedge}dy_{3}+x_{1}\otimes
dx_{3}{\wedge}dy_{2}-y_{1}\otimes dx_{2}{\wedge}dy_{3}\end{array}$

######  Comment.

Since we know that ${\mathfrak{br}}(2;a)$ admits a 3-parameter family of
deformations ([GL4]), three of these five cocycles (recall also the ones of
degrees 3 and 6) must be integrable. By symmetry the algebras obtained from
the cocycles of of degrees $\pm 3$ are isomorphic and so are the algebras
obtained from the cocycles of of degrees $\pm 6$. Anyway, the $\geq 3$
cocycles in the above case are expected, so all cocycles are integrable, as in
[DK]. Contrariwise, the next result is not expected; and we find it a bit too
much (are all these cocycle integrable?!), as in [Ch, Vi]. The application of
the group $\mathop{\mathrm{{Aut\/}}}\nolimits({\mathfrak{g}})$ by Kuznetsov
and Chebochko [KuCh, Ch1] makes the abundance of cocycles not so scary since
only the $\mathop{\mathrm{{Aut\/}}}\nolimits({\mathfrak{g}})$-orbits matter.

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{br}}(3)$, we consider the Cartan matrix

(16) $\begin{pmatrix}2&-1&0\\\ -1&2&-1\\\
0&-1&{\bar{0}}\end{pmatrix}\quad\text{and the basis
}\;\tiny\begin{array}[]{l}x_{1},\;x_{2},\;x_{3};\\\
{}x_{4}=[x_{1},x_{2}],\quad x_{5}=[x_{2},x_{3}];\\\
{}x_{6}=[x_{3},[x_{1},x_{2}]],\quad x_{7}=[x_{3},[x_{2},x_{3}]];\\\
{}x_{8}=[x_{3},[x_{3},[x_{1},x_{2}]]];\\\
{}x_{9}=[[x_{2},x_{3}],[x_{3},[x_{1},x_{2}]]];\\\
{}x_{10}=[[x_{3},[x_{1},x_{2}]],[x_{3},[x_{2},x_{3}]]];\\\
{}x_{11}=[[x_{3},[x_{2},x_{3}]],[x_{3},[x_{3},[x_{1},x_{2}]]]];\\\
{}x_{12}=[[x_{3},[x_{2},x_{3}]],[[x_{2},x_{3}],[x_{3},[x_{1},x_{2}]]]];\\\
{}x_{13}=[[x_{3},[x_{3},[x_{1},x_{2}]]],[[x_{2},x_{3}],[x_{3},[x_{1},x_{2}]]]].\end{array}$

Then $H^{2}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycles
$(\ref{eqbr3})$.

(17) $\tiny\begin{array}[]{ll}\deg=-18:&2\,y_{3}\otimes
dx_{12}{\wedge}dx_{13}+2\,y_{5}\otimes dx_{11}{\wedge}dx_{13}+2\,y_{6}\otimes
dx_{11}{\wedge}dx_{12}+y_{7}\otimes dx_{10}{\wedge}dx_{13}+y_{8}\otimes
dx_{10}{\wedge}dx_{12}+\\\ &y_{9}\otimes
dx_{10}{\wedge}dx_{11}+2\,y_{10}\otimes dx_{7}{\wedge}dx_{13}+y_{10}\otimes
dx_{8}{\wedge}dx_{12}+2\,y_{10}\otimes dx_{9}{\wedge}dx_{11}+2\,y_{11}\otimes
dx_{5}{\wedge}dx_{13}+\\\ &y_{11}\otimes
dx_{6}{\wedge}dx_{12}+2\,y_{11}\otimes dx_{9}{\wedge}dx_{10}+2\,y_{12}\otimes
dx_{3}{\wedge}dx_{13}+y_{12}\otimes dx_{6}{\wedge}dx_{11}+2\,y_{12}\otimes
dx_{8}{\wedge}dx_{10}\\\ &+2\,y_{13}\otimes
dx_{3}{\wedge}dx_{12}+y_{13}\otimes dx_{5}{\wedge}dx_{11}+2\,y_{13}\otimes
dx_{7}{\wedge}dx_{10};\\\ &\\\ \deg=-9:&2\,x_{3}\otimes
dx_{1}{\wedge}dx_{13}+x_{5}\otimes dx_{4}{\wedge}dx_{13}+x_{7}\otimes
dx_{6}{\wedge}dx_{13}+y_{1}\otimes dx_{4}{\wedge}dx_{10}+2\,y_{1}\otimes
dx_{6}{\wedge}dx_{9}+\\\ &2\,y_{1}\otimes dx_{13}{\wedge}dy_{3}+y_{4}\otimes
dx_{1}{\wedge}dx_{10}+y_{4}\otimes dx_{6}{\wedge}dx_{8}+y_{4}\otimes
dx_{13}{\wedge}dy_{5}+y_{6}\otimes dx_{1}{\wedge}dx_{9}+\\\ &y_{6}\otimes
dx_{4}{\wedge}dx_{8}+2\,y_{6}\otimes dx_{13}{\wedge}dy_{7}+2\,y_{8}\otimes
dx_{4}{\wedge}dx_{6}+y_{9}\otimes dx_{1}{\wedge}dx_{6}+2\,y_{10}\otimes
dx_{1}{\wedge}dx_{4}\\\ &+y_{13}\otimes dx_{1}{\wedge}dy_{3}+y_{13}\otimes
dx_{4}{\wedge}dy_{5}+y_{13}\otimes dx_{6}{\wedge}dy_{7};\\\ &\\\
\deg=-6:&2\,x_{1}\otimes dx_{2}{\wedge}dx_{10}+2\,x_{1}\otimes
dx_{5}{\wedge}dx_{9}+2\,x_{1}\otimes dx_{12}{\wedge}dy_{3}+x_{3}\otimes
dx_{12}{\wedge}dy_{1}+2\,x_{6}\otimes dx_{2}{\wedge}dx_{12}+\\\ &x_{8}\otimes
dx_{5}{\wedge}dx_{12}+2\,y_{2}\otimes dx_{5}{\wedge}dx_{7}+2\,y_{2}\otimes
dx_{10}{\wedge}dy_{1}+2\,y_{2}\otimes dx_{12}{\wedge}dy_{6}+2\,y_{5}\otimes
dx_{2}{\wedge}dx_{7}+\\\ &y_{5}\otimes dx_{9}{\wedge}dy_{1}+2\,y_{5}\otimes
dx_{12}{\wedge}dy_{8}+y_{7}\otimes dx_{2}{\wedge}dx_{5}+2\,y_{9}\otimes
dx_{5}{\wedge}dy_{1}+2\,y_{10}\otimes dx_{2}{\wedge}dy_{1}\\\
&+2\,y_{12}\otimes dx_{2}{\wedge}dy_{6}+y_{12}\otimes
dx_{5}{\wedge}dy_{8}+2\,y_{12}\otimes dy_{1}{\wedge}dy_{3};\\\ &\\\
\deg=-3:&2\,x_{2}\otimes dx_{3}{\wedge}dx_{7}+x_{2}\otimes
dx_{10}{\wedge}dy_{4}+2\,x_{2}\otimes dx_{11}{\wedge}dy_{6}+x_{4}\otimes
dx_{3}{\wedge}dx_{8}+x_{4}\otimes dx_{10}{\wedge}dy_{2}+\\\ &x_{4}\otimes
dx_{11}{\wedge}dy_{5}+2\,x_{5}\otimes dx_{11}{\wedge}dy_{4}+x_{6}\otimes
dx_{11}{\wedge}dy_{2}+x_{9}\otimes dx_{3}{\wedge}dx_{11}+2\,y_{3}\otimes
dx_{7}{\wedge}dy_{2}+\\\ &2\,y_{3}\otimes dx_{8}{\wedge}dy_{4}+2\,y_{3}\otimes
dx_{11}{\wedge}dy_{9}+2\,y_{7}\otimes dx_{3}{\wedge}dy_{2}+y_{8}\otimes
dx_{3}{\wedge}dy_{4}+y_{10}\otimes dy_{2}{\wedge}dy_{4}\\\ &+y_{11}\otimes
dx_{3}{\wedge}dy_{9}+y_{11}\otimes dy_{2}{\wedge}dy_{6}+y_{11}\otimes
dy_{4}{\wedge}dy_{5};\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{brj}}(2;3)$, we consider the Cartan matrix

(18) $\begin{pmatrix}0&-1\\\ -2&1\end{pmatrix}\quad\begin{matrix}\text{and the
basis }\\\ \text{even $|$ odd
}\end{matrix}\;\tiny\begin{array}[]{l}\;|\;x_{1},\;x_{2},\\\
x_{3}=\left[x_{1},\,x_{2}\right]\;|\\\ x_{4}=\left[x_{2},\,x_{2}\right],\;|\\\
\;|\;x_{5}=\left[x_{2},\,\left[x_{1},\,x_{2}\right]\right]\\\
x_{6}=\left[\left[x_{1},\,x_{2}\right],\,\left[x_{2},\,x_{2}\right]\right],\;|\\\
\;|\;x_{7}=\left[\left[x_{2},\,x_{2}\right],\,\left[x_{2},\,\left[x_{1},\,x_{2}\right]\right]\right]\\\
x_{8}=\left[\left[x_{1},\,x_{2}\right],\,\left[\left[x_{1},\,x_{2}\right],\,\left[x_{2},\,x_{2}\right]\right]\right]\;|\end{array}$

Then $H^{2}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycles

(19) $\tiny\begin{array}[]{ll}\deg=-12:&2\,y_{2}\otimes
dx_{7}{\wedge}dx_{8}+y_{3}\otimes dx_{7}{\wedge}dx_{7}+y_{4}\otimes
dx_{6}{\wedge}dx_{8}+y_{5}\otimes dx_{6}{\wedge}dx_{7}+2\,y_{6}\otimes
dx_{4}{\wedge}dx_{8}+\\\ &2\,y_{6}\otimes dx_{5}{\wedge}dx_{7}+y_{7}\otimes
dx_{2}{\wedge}dx_{8}+2\,y_{7}\otimes dx_{3}{\wedge}dx_{7}+2\,y_{7}\otimes
dx_{5}{\wedge}dx_{6}+2\,y_{8}\otimes dx_{2}{\wedge}dx_{7}+\\\ &2\,y_{8}\otimes
dx_{4}{\wedge}dx_{6}\\\ &\\\ \deg=-6:&x_{2}\otimes
dx_{1}{\wedge}dx_{8}+2\,x_{4}\otimes dx_{3}{\wedge}dx_{8}+2\,y_{1}\otimes
dx_{1}{\wedge}dx_{6}+2\,y_{1}\otimes dx_{3}{\wedge}dx_{5}+y_{1}\otimes
dx_{8}{\wedge}dy_{2}+\\\ &2\,y_{3}\otimes dx_{1}{\wedge}dx_{5}+2\,y_{3}\otimes
dx_{8}{\wedge}dy_{4}+y_{5}\otimes dx_{1}{\wedge}dx_{3}+y_{6}\otimes
dx_{1}{\wedge}dx_{1}+2\,y_{8}\otimes dx_{1}{\wedge}dy_{2}+\\\ &2\,y_{8}\otimes
dx_{3}{\wedge}dy_{4}\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{brj}}(2;5)$, we consider the Cartan matrix

(20) $\begin{pmatrix}0&-1\\\ -2&1\end{pmatrix}\quad\begin{matrix}\text{and the
basis }\\\ \text{even $|$ odd
}\end{matrix}\;\tiny\begin{array}[]{l}\;|\;x_{1},\;x_{2},\\\
x_{3}=\left[x_{1},\,x_{2}\right],\;x_{4}=\left[x_{2},\,x_{2}\right],\;|\\\
\;|\;x_{5}=\left[x_{2},\,\left[x_{1},\,x_{2}\right]\right]\\\
x_{6}=\left[\left[x_{1},\,x_{2}\right],\,\left[x_{2},\,x_{2}\right]\right],\;|\\\
\;|\;x_{7}=\left[\left[x_{1},\,x_{2}\right],\,\left[x_{2},\,\left[x_{1},\,x_{2}\right]\right]\right],\;x_{8}=\left[\left[x_{2},\,x_{2}\right],\,\left[x_{2},\,\left[x_{1},\,x_{2}\right]\right]\right],\\\
x_{9}=\left[\left[x_{1},\,x_{2}\right],\,\left[\left[x_{1},\,x_{2}\right],\,\left[x_{2},\,x_{2}\right]\right]\right]\;|\\\
\;|\;x_{10}=\left[\left[x_{2},\,\left[x_{1},\,x_{2}\right]\right],\,\left[\left[x_{1},\,x_{2}\right],\,\left[x_{2},\,x_{2}\right]\right]\right]\end{array}$

Then $H^{2}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycles

(21) $\tiny\begin{array}[]{ll}\underline{\deg=-15}&2\,y_{1}\otimes
dx_{10}\wedge dx_{10}+3\,y_{3}\otimes dx_{9}\wedge dx_{10}+y_{5}\otimes
dx_{7}\wedge dx_{10}+3\,y_{6}\otimes dx_{7}\wedge dx_{9}+3\,y_{7}\otimes
dx_{5}\wedge dx_{10}+\\\ &4\,y_{7}\otimes dx_{6}\wedge dx_{9}+y_{7}\otimes
dx_{7}\wedge dx_{8}+2\,y_{8}\otimes dx_{7}\wedge dx_{7}+4\,y_{9}\otimes
dx_{3}\wedge dx_{10}+y_{9}\otimes dx_{6}\wedge dx_{7}+\\\ &4\,y_{10}\otimes
dx_{1}\wedge dx_{10}+4\,y_{10}\otimes dx_{3}\wedge dx_{9}+2\,y_{10}\otimes
dx_{5}\wedge dx_{7}\\\ &\\\ \underline{\deg=-5}:&3\,x_{1}\otimes
dx_{2}{\wedge}dx_{8}+4\,x_{1}\otimes dx_{4}{\wedge}dx_{6}+3\,x_{1}\otimes
dx_{10}{\wedge}dy_{1}+4\,x_{3}\otimes dx_{4}{\wedge}dx_{8}+3\,x_{7}\otimes
dx_{8}){\wedge}dx_{8}+\\\ &y_{2}\otimes dx_{8}{\wedge}dy_{1}+4\,y_{4}\otimes
dx_{6}{\wedge}dy_{1}+2\,y_{4}\otimes dx_{8}{\wedge}dy_{3}+2\,y_{6}\otimes
dx_{4}{\wedge}dy_{1}+2\,y_{8}\otimes dx_{2}{\wedge}dy_{1}+\\\ &3\,y_{8}\otimes
dx_{4}{\wedge}dy_{3}+y_{8}\otimes dx_{8}{\wedge}dy_{7}+y_{10}\otimes
dy_{1}{\wedge}dy_{1}\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{g}}(1,6)$, we consider the Cartan matrix

(22) $\begin{pmatrix}2&-1&0\\\ -1&1&-1\\\
0&-1&0\end{pmatrix}\quad\begin{matrix}\text{and the basis }\\\ \text{even $|$
odd }\end{matrix}\;\tiny\begin{array}[]{l}x_{1},\;|\;x_{2},\;x_{3}\\\
x_{5}=\left[x_{2},\,x_{2}\right],\;|\;x_{4}=\left[x_{1},\,x_{2}\right],\\\
x_{6}=\left[x_{2},\,x_{3}\right],\;x_{7}=\left[x_{2},\,\left[x_{1},\,x_{2}\right]\right]\;|\\\
x_{8}=\left[x_{3},\,\left[x_{1},\,x_{2}\right]\right],\;|\;x_{9}=\left[x_{3},\,\left[x_{2},\,x_{2}\right]\right]\\\
x_{10}=\left[\left[x_{1},\,x_{2}\right],\,\left[x_{1},\,x_{2}\right]\right],\;|\;x_{11}=\left[\left[x_{1},\,x_{2}\right],\,\left[x_{2},\,x_{3}\right]\right]\\\
\;|\;x_{12}=\left[\left[x_{1},\,x_{2}\right],\,\left[x_{3},\,\left[x_{1},\,x_{2}\right]\right]\right]\\\
x_{13}=\left[\left[x_{2},\,x_{3}\right],\,\left[x_{2},\,\left[x_{1},\,x_{2}\right]\right]\right]\;|\\\
x_{14}=\left[\left[x_{2},\,\left[x_{1},\,x_{2}\right]\right],\,\left[x_{3},\,\left[x_{1},\,x_{2}\right]\right]\right]\;|\\\
\;|\;x_{15}=\left[\left[x_{3},\,\left[x_{2},\,x_{2}\right]\right],\,\left[\left[x_{1},\,x_{2}\right],\,\left[x_{1},\,x_{2}\right]\right]\right]\\\
x_{16}=\left[\left[x_{3},\,\left[x_{2},\,x_{2}\right]\right],\,\left[\left[x_{1},\,x_{2}\right],\,\left[x_{3},\,\left[x_{1},\,x_{2}\right]\right]\right]\right]\;|\end{array}$

Then $H^{2}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycles
$(\ref{eq1,3})$.

(23) $\tiny\begin{array}[]{ll}\underline{\deg=-12}:&2\,x_{2}\otimes
dx_{13}{\wedge}dx_{16}+2\,x_{4}\otimes dx_{14}{\wedge}dx_{16}+y_{2}\otimes
dx_{8}{\wedge}dx_{16}+2\,y_{3}\otimes dx_{13}{\wedge}dx_{14}+y_{4}\otimes
dx_{6}{\wedge}dx_{16}+\\\ &y_{6}\otimes dx_{4}{\wedge}dx_{16}+y_{6}\otimes
dx_{8}{\wedge}dx_{15}+2\,y_{6}\otimes dx_{11}{\wedge}dx_{14}+2\,y_{6}\otimes
dx_{12}{\wedge}dx_{13}+y_{8}\otimes dx_{2}{\wedge}dx_{16}+\\\ &y_{8}\otimes
dx_{6}{\wedge}dx_{15}+y_{8}\otimes dx_{9}{\wedge}dx_{14}+y_{8}\otimes
dx_{11}{\wedge}dx_{13}+y_{9}\otimes dx_{8}{\wedge}dx_{14}+y_{11}\otimes
dx_{6}{\wedge}dx_{14}\\\ &+y_{11}\otimes dx_{8}{\wedge}dx_{13}+y_{12}\otimes
dx_{6}{\wedge}dx_{13}+2\,y_{13}\otimes dx_{3}{\wedge}dx_{14}+2\,y_{13}\otimes
dx_{6}{\wedge}dx_{12}+2\,y_{13}\otimes dx_{8}{\wedge}dx_{11}+\\\
&y_{13}\otimes dx_{16}{\wedge}dy_{2}+2\,y_{14}\otimes
dx_{3}{\wedge}dx_{13}+y_{14}\otimes dx_{6}{\wedge}dx_{11}+y_{14}\otimes
dx_{8}{\wedge}dx_{9}+y_{14}\otimes dx_{16}{\wedge}dy_{4}+\\\ &2\,y_{15}\otimes
dx_{6}{\wedge}dx_{8}+y_{16}\otimes dx_{2}{\wedge}dx_{8}+2\,y_{16}\otimes
dx_{4}{\wedge}dx_{6}+y_{16}\otimes dx_{13}{\wedge}dy_{2}+2\,y_{16}\otimes
dx_{14}{\wedge}dy_{4};\\\ \underline{\deg=-6}:&x_{2}\otimes\left(dx_{7}\wedge
dx_{10}\right)+x_{3}\otimes\left(dx_{1}\wedge
dx_{14}\right)+x_{3}\otimes\left(dx_{4}\wedge
dx_{12}\right)+x_{3}\otimes\left(dx_{8}\wedge dx_{10}\right)\text{+
}2\,x_{6}\otimes\left(dx_{1}\wedge dx_{15}\right)+\\\
&2\,x_{6}\otimes\left(dx_{7}\wedge
dx_{12}\right)+x_{8}\otimes\left(dx_{10}\wedge
dx_{12}\right)+x_{9}\otimes\left(dx_{4}\wedge dx_{15}\right)\text{+
}2\,x_{9}\otimes\left(dx_{7}\wedge dx_{14}\right)\text{+
}2\,x_{9}\otimes\left(dx_{10}\wedge dx_{13}\right)\\\
&+x_{13}\otimes\left(dx_{10}\wedge dx_{15}\right)\text{+
}2\,y_{1}\otimes\left(dx_{2}\wedge dx_{10}\right)\text{+
}2\,y_{1}\otimes\left(dx_{4}\wedge
dx_{7}\right)+y_{1}\otimes\left(dx_{14}\wedge
dy_{3}\right)+y_{1}\otimes\left(dx_{15}\wedge dy_{6}\right)\\\
&+y_{2}\otimes\left(dx_{1}\wedge dx_{10}\right)\text{+
}2\,y_{4}\otimes\left(dx_{1}\wedge
dx_{7}\right)+y_{4}\otimes\left(dx_{12}\wedge
dy_{3}\right)+y_{4}\otimes\left(dx_{15}\wedge
dy_{9}\right)+y_{7}\otimes\left(dx_{1}\wedge dx_{4}\right)\\\
&+y_{7}\otimes\left(dx_{10}\wedge
dy_{2}\right)+y_{7}\otimes\left(dx_{12}\wedge dy_{6}\right)\text{+
}2\,y_{7}\otimes\left(dx_{14}\wedge dy_{9}\right)\text{+
}2\,y_{8}\otimes\left(dx_{10}\wedge
dy_{3}\right)+y_{10}\otimes\left(dx_{1}\wedge dx_{2}\right)\\\ &\text{+
}2\,y_{10}\otimes\left(dx_{7}\wedge dy_{2}\right)\text{+
}2\,y_{10}\otimes\left(dx_{8}\wedge
dy_{3}\right)+y_{10}\otimes\left(dx_{12}\wedge dy_{8}\right)\text{+
}2\,y_{10}\otimes\left(dx_{13}\wedge
dy_{9}\right)+y_{10}\otimes\left(dx_{15}\wedge dy_{13}\right)\\\ &\text{+
}2\,y_{12}\otimes\left(dx_{4}\wedge dy_{3}\right)\text{+
}2\,y_{12}\otimes\left(dx_{7}\wedge dy_{6}\right)\text{+
}2\,y_{12}\otimes\left(dx_{10}\wedge dy_{8}\right)\text{+
}2\,y_{13}\otimes\left(dx_{10}\wedge dy_{9}\right)\text{+
}2\,y_{14}\otimes\left(dx_{1}\wedge dy_{3}\right)\\\
&+y_{14}\otimes\left(dx_{7}\wedge dy_{9}\right)\text{+
}2\,y_{15}\otimes\left(dx_{1}\wedge dy_{6}\right)\text{+
}2\,y_{15}\otimes\left(dx_{4}\wedge dy_{9}\right)\text{+
}2\,y_{15}\otimes\left(dx_{10}\wedge dy_{13}\right);\\\
\underline{\deg=-3}:&2\,x_{1}\otimes\left(dx_{2}\wedge
dx_{7}\right)+2\,x_{1}\otimes\left(dx_{4}\wedge
dx_{5}\right)+2\,x_{1}\otimes\left(dx_{13}\wedge
dy_{3}\right)+2\,x_{1}\otimes\left(dx_{15}\wedge
dy_{8}\right)+x_{3}\otimes\left(dx_{2}\wedge dx_{9}\right)\\\
&+2\,x_{3}\otimes\left(dx_{5}\wedge
dx_{6}\right)+x_{3}\otimes\left(dx_{13}\wedge
dy_{1}\right)+2\,x_{4}\otimes\left(dx_{5}\wedge
dx_{7}\right)+x_{6}\otimes\left(dx_{5}\wedge dx_{9}\right)\\\
&+2\,x_{8}\otimes\left(dx_{7}\wedge
dx_{9}\right)+2\,x_{8}\otimes\left(dx_{15}\wedge
dy_{1}\right)+x_{12}\otimes\left(dx_{2}\wedge
dx_{15}\right)+x_{12}\otimes\left(dx_{5}\wedge
dx_{14}\right)+x_{12}\otimes\left(dx_{7}\wedge dx_{13}\right)\\\ &\text{+
}2\,x_{14}\otimes\left(dx_{5}\wedge dx_{15}\right)\text{+
}2\,y_{2}\otimes\left(dx_{7}\wedge
dy_{1}\right)+2\,y_{2}\otimes\left(dx_{9}\wedge
dy_{3}\right)+2\,y_{2}\otimes\left(dx_{15}\wedge
dy_{12}\right)+y_{4}\otimes\left(dx_{5}\wedge dy_{1}\right)\\\
&+y_{5}\otimes\left(dx_{4}\wedge
dy_{1}\right)+2\,y_{5}\otimes\left(dx_{6}\wedge
dy_{3}\right)+y_{5}\otimes\left(dx_{7}\wedge
dy_{4}\right)+2\,y_{5}\otimes\left(dx_{9}\wedge
dy_{6}\right)+y_{5}\otimes\left(dx_{14}\wedge dy_{12}\right)\\\
&+y_{5}\otimes\left(dx_{15}\wedge
dy_{14}\right)+y_{6}\otimes\left(dx_{5}\wedge
dy_{3}\right)+y_{7}\otimes\left(dx_{2}\wedge
dy_{1}\right)+2\,y_{7}\otimes\left(dx_{5}\wedge
dy_{4}\right)+2\,y_{7}\otimes\left(dx_{9}\wedge dy_{8}\right)\\\
&+y_{7}\otimes\left(dx_{13}\wedge dy_{12}\right)\text{+
}2\,y_{9}\otimes\left(dx_{2}\wedge
dy_{3}\right)+y_{9}\otimes\left(dx_{5}\wedge
dy_{6}\right)+y_{9}\otimes\left(dx_{7}\wedge
dy_{8}\right)+y_{13}\otimes\left(dx_{7}\wedge dy_{12}\right)\\\ &\text{+
}2\,y_{13}\otimes\left(dy_{1}\wedge
dy_{3}\right)+2\,y_{14}\otimes\left(dx_{5}\wedge dy_{12}\right)\text{+
}2\,y_{15}\otimes\left(dx_{2}\wedge dy_{12}\right)\text{+
}2\,y_{15}\otimes\left(dx_{5}\wedge
dy_{14}\right)+y_{15}\otimes\left(dy_{1}\wedge dy_{8}\right).\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{g}}(2,3)$, we consider the Cartan matrix

(24) $\begin{pmatrix}2&-1&-1\\\ -1&2&-1\\\
-1&-1&0\end{pmatrix}\quad\begin{matrix}\text{and the basis }\\\ \text{even $|$
odd }\end{matrix}\;\tiny\begin{array}[]{l}x_{1},\;x_{2},\;|\;x_{3}\\\
x_{4}=\left[x_{1},\,x_{2}\right],\;|\;x_{5}=\left[x_{1},\,x_{3}\right]\\\
\;|\;x_{6}=\left[x_{2},\,x_{3}\right],\;x_{7}=\left[x_{3},\,\left[x_{1},\,x_{2}\right]\right],\\\
\;|\;x_{8}=\left[\left[x_{1},\,x_{2}\right],\,\left[x_{1},\,x_{3}\right]\right],\;x_{9}=\left[\left[x_{1},\,x_{2}\right],\,\left[x_{2},\,x_{3}\right]\right]\\\
\;|\;x_{10}=\left[\left[x_{1},\,x_{2}\right],\,\left[x_{3},\,\left[x_{1},\,x_{2}\right]\right]\right]\\\
x_{11}=\left[\left[x_{2},\,x_{3}\right],\,\left[\left[x_{1},\,x_{2}\right],\,\left[x_{1},\,x_{3}\right]\right]\right]\;|\end{array}$

Then $H^{2}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycles
$(\ref{eqg2,3})$.

(25) $\tiny\begin{array}[]{ll}\underline{\deg=-9}:&2\,x_{2}\otimes
dx_{9}{\wedge}dx_{11}+2\,x_{4}\otimes dx_{10}{\wedge}dx_{11}+y_{2}\otimes
dx_{5}{\wedge}dx_{11}+y_{3}\otimes dx_{4}{\wedge}dx_{11}+2\,y_{3}\otimes
dx_{7}{\wedge}dx_{10}+\\\ &y_{3}\otimes dx_{8}{\wedge}dx_{9}+y_{4}\otimes
dx_{3}{\wedge}dx_{11}+y_{5}\otimes dx_{2}{\wedge}dx_{11}+2\,y_{5}\otimes
dx_{6}{\wedge}dx_{10}+y_{5}\otimes dx_{7}{\wedge}dx_{9}+\\\ &2\,y_{6}\otimes
dx_{5}){\wedge}dx_{10}+y_{7}\otimes dx_{3}{\wedge}dx_{10}+y_{7}\otimes
dx_{5}{\wedge}dx_{9}+2\,y_{8}\otimes dx_{3}{\wedge}dx_{9}+2\,y_{9}\otimes
dx_{3}{\wedge}dx_{8}+\\\ &y_{9}\otimes dx_{5}{\wedge}dx_{7}+y_{9}\otimes
dx_{11}{\wedge}dy_{2}+2\,y_{10}\otimes dx_{3}{\wedge}dx_{7}+y_{10}\otimes
dx_{5}{\wedge}dx_{6}+y_{10}\otimes dx_{11}{\wedge}dy_{4}+\\\ &2\,y_{11}\otimes
dx_{2}{\wedge}dx_{5}+y_{11}\otimes dx_{3}{\wedge}dx_{4}+y_{11}\otimes
dx_{9}{\wedge}dy_{2}+2\,y_{11}\otimes dx_{10}{\wedge}dy_{4}\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{g}}(3,3)$ and
${\mathfrak{g}}={\mathfrak{g}}(4,3)$, we have
$H^{2}({\mathfrak{g}};{\mathfrak{g}})=0$.

######  Lemma.

The “classical” (serial) simple Lie superalgebras of rank $=2$ and $3$ and
with Cartan matrix are rigid if $p=3$ and $5$, except for
${\mathfrak{osp}}(4|2)$. The corresponding cocycles are (26) for $p\geq 5$ and
$p=0$:

(26) $\tiny\begin{array}[]{l}2h_{1}\otimes dx_{2}\wedge dy_{2}+3h_{1}\otimes
dx_{3}\wedge dy_{3}+2h_{1}\otimes dx_{4}\wedge dy_{4}+h_{1}\otimes
dx_{6}\wedge dy_{6}+3h_{1}\otimes dx_{7}\wedge dy_{7}+\\\ h_{2}\otimes
dx_{3}\wedge dy_{3}+4h_{2}\otimes dx_{5}\wedge dy_{5}+h_{2}\otimes
dx_{6}\wedge dy_{6}+h_{2}\otimes dx_{7}\wedge dy_{7}+4x_{1}\otimes
dh_{3}\wedge dx_{1}+\\\ x_{1}\otimes dx_{4}\wedge dy_{2}+x_{1}\otimes
dx_{6}\wedge dy_{5}+2x_{2}\otimes dh_{2}\wedge dx_{2}+x_{2}\otimes
dh_{3}\wedge dx_{2}+3x_{3}\otimes dh_{2}\wedge dx_{3}+\\\ 2x_{4}\otimes
dh_{2}\wedge dx_{4}+x_{5}\otimes dh_{3}\wedge dx_{5}+3x_{5}\otimes
dx_{2}\wedge dx_{3}+2x_{6}\otimes dx_{3}\wedge dx_{4}+2x_{7}\otimes
dh_{2}\wedge dx_{7}+\\\ x_{7}\otimes dh_{3}\wedge dx_{7}+3x_{7}\otimes
dx_{2}\wedge dx_{6}+3x_{7}\otimes dx_{4}\wedge dx_{5}+y_{1}\otimes
dh_{3}\wedge dy_{1}+4y_{1}\otimes dx_{2}\wedge dy_{4}+\\\ 4y_{1}\otimes
dx_{5}\wedge dy_{6}+3y_{2}\otimes dh_{2}\wedge dy_{2}+4y_{2}\otimes
dh_{3}\wedge dy_{2}+3y_{2}\otimes dx_{3}\wedge dy_{5}+2y_{2}\otimes
dx_{6}\wedge dy_{7}+\\\ 2y_{3}\otimes dh_{2}\wedge dy_{3}+2y_{3}\otimes
dx_{2}\wedge dy_{5}+2y_{3}\otimes dx_{4}\wedge dy_{6}+3y_{4}\otimes
dh_{2}\wedge dy_{4}+2y_{4}\otimes dx_{3}\wedge dy_{6}+\\\ 3y_{4}\otimes
dx_{5}\wedge dy_{7}+4y_{5}\otimes dh_{3}\wedge dy_{5}+3y_{5}\otimes
dx_{4}\wedge dy_{7}+2y_{6}\otimes dx_{2}\wedge dy_{7}+3y_{7}\otimes
dh_{2}\wedge dy_{7}+\\\ 4y_{7}\otimes dh_{3}\wedge dy_{7}\end{array}$

and (27) for $p=3$:

(27) $\tiny\begin{array}[]{l}h_{1}\otimes dx_{2}\wedge dy_{2}+h_{1}\otimes
dx_{3}\wedge dy_{3}+h_{1}\otimes dx_{4}\wedge dy_{4}+h_{1}\otimes dx_{5}\wedge
dy_{5}+\\\ h_{1}\otimes dx_{7}\wedge dy_{7}+2h_{2}\otimes dx_{3}\wedge
dy_{3}+h_{2}\otimes dx_{7}\wedge dy_{7}+x_{1}\otimes dh_{3}\wedge dx_{1}+\\\
x_{1}\otimes dx_{4}\wedge dy_{2}+x_{1}\otimes dx_{6}\wedge dy_{5}+x_{2}\otimes
dh_{2}\wedge dx_{2}+2x_{2}\otimes dh_{3}\wedge dx_{2}+\\\ 2x_{2}\otimes
dx_{5}\wedge dy_{3}+x_{2}\otimes dx_{7}\wedge dy_{6}+2x_{3}\otimes
dh_{2}\wedge dx_{3}+x_{4}\otimes dh_{2}\wedge dx_{4}+\\\ x_{4}\otimes
dx_{6}\wedge dy_{3}+2x_{4}\otimes dx_{7}\wedge dy_{5}+2x_{5}\otimes
dh_{3}\wedge dx_{5}+2x_{5}\otimes dx_{2}\wedge dx_{3}+\\\ x_{6}\otimes
dx_{3}\wedge dx_{4}+x_{7}\otimes dh_{2}\wedge dx_{7}+2x_{7}\otimes
dh_{3}\wedge dx_{7}+2x_{7}\otimes dx_{2}\wedge dx_{6}+\\\ 2x_{7}\otimes
dx_{4}\wedge dx_{5}+2y_{1}\otimes dh_{3}\wedge dy_{1}+2y_{1}\otimes
dx_{2}\wedge dy_{4}+2y_{1}\otimes dx_{5}\wedge dy_{6}+\\\ 2y_{2}\otimes
dh_{2}\wedge dy_{2}+y_{2}\otimes dh_{3}\wedge dy_{2}+y_{2}\otimes dx_{3}\wedge
dy_{5}+y_{3}\otimes dh_{2}\wedge dy_{3}+\\\ y_{3}\otimes dx_{2}\wedge
dy_{5}+y_{3}\otimes dx_{4}\wedge dy_{6}+2y_{4}\otimes dh_{2}\wedge
dy_{4}+2y_{4}\otimes dx_{3}\wedge dy_{6}+\\\ y_{5}\otimes dh_{3}\wedge
dy_{5}+2y_{5}\otimes dx_{4}\wedge dy_{7}+y_{6}\otimes dx_{2}\wedge
dy_{7}+2y_{7}\otimes dh_{2}\wedge dy_{7}+y_{7}\otimes dh_{3}\wedge
dy_{7}\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{psq}}(3)$, we have
$H^{2}({\mathfrak{g}};{\mathfrak{g}})=0$ for $p=5$.

For $p=3$, when ${\mathfrak{g}}={\mathfrak{psq}}(3)$ is not simple, we take
the simple algebra ${\mathfrak{g}}^{\prime}={\mathfrak{psq}}^{(1)}(3)$ and
$H^{2}({\mathfrak{g}}^{\prime};{\mathfrak{g}}^{\prime})$ is spanned by the
$2$-cocycle

(28)
$\tiny\begin{array}[]{ll}\underline{\deg=0}:&2\,a_{2,2}\otimes\text{da}_{2,2}\wedge\text{db}_{2}+2\,a_{2,2}\otimes\text{da}_{3,3}\wedge\text{db}_{2}+2\,a_{2,2}\otimes\text{da}_{1,2}\wedge\text{db}_{2,1}+a_{2,2}\otimes\text{da}_{1,3}\wedge\text{db}_{3,1}+a_{2,2}\otimes\text{da}_{3,2}\wedge\text{db}_{2,3}+\\\
&a_{2,2}\otimes\text{da}_{3,1}\wedge\text{db}_{1,3}+2\,a_{3,3}\otimes\text{da}_{2,2}\wedge\text{db}_{1}+2\,a_{3,3}\otimes\text{da}_{2,2}\wedge\text{db}_{2}+2\,a_{3,3}\otimes\text{da}_{3,3}\wedge\text{db}_{1}+2\,a_{3,3}\otimes\text{da}_{3,3}\wedge\text{db}_{2}+\\\
&a_{3,3}\otimes\text{da}_{2,1}\wedge\text{db}_{1,2}+2\,a_{3,3}\otimes\text{da}_{3,2}\wedge\text{db}_{2,3}+a_{3,3}\otimes\text{da}_{3,1}\wedge\text{db}_{1,3}+2\,a_{1,2}\otimes\text{da}_{1,2}\wedge\text{db}_{2})+2\,a_{1,2}\otimes\text{da}_{3,2}\wedge\text{db}_{1,3}\\\
&+a_{2,3}\otimes\text{da}_{2,3}\wedge\text{db}_{2}+a_{1,3}\otimes\text{da}_{2,2}\wedge\text{db}_{1,3}+a_{1,3}\otimes\text{da}_{3,3}\wedge\text{db}_{1,3}+a_{1,3}\otimes\text{da}_{2,3}\wedge\text{db}_{1,2}+a_{1,3}\otimes\text{da}_{1,3}\wedge\text{db}_{2}+\\\
&2\,a_{2,1}\otimes\text{da}_{2,1}\wedge\text{db}_{1}+a_{3,2}\otimes\text{da}_{3,2}\wedge\text{db}_{1}+a_{3,1}\otimes\text{da}_{2,2}\wedge\text{db}_{3,1}+a_{3,1}\otimes\text{da}_{3,3}\wedge\text{db}_{3,1}+2\,a_{3,1}\otimes\text{da}_{3,1}\wedge\text{db}_{1}+\\\
&a_{3,1}\otimes\text{da}_{3,1}\wedge\text{db}_{2}+2\,b_{1}\otimes\text{db}_{1}\wedge\text{db}_{1}+b_{1}\otimes\text{db}_{1,3}\wedge\text{db}_{3,1}+b_{2}\otimes\text{db}_{2}\wedge\text{db}_{2}+b_{1,2}\otimes\text{da}_{2,2}\wedge\text{da}_{1,2}\\\
&+b_{1,2}\otimes\text{da}_{3,3}\wedge\text{da}_{1,2}+b_{1,2}\otimes\text{da}_{1,3}\wedge\text{da}_{3,2}+b_{1,2}\otimes\text{db}_{1,3}\wedge\text{db}_{3,2}+2\,b_{2,3}\otimes\text{da}_{2,2}\wedge\text{da}_{2,3}+2\,b_{2,3}\otimes\text{da}_{3,3}\wedge\text{da}_{2,3}+\\\
&2\,b_{2,3}\otimes\text{da}_{1,3}\wedge\text{da}_{2,1}+b_{2,3}\otimes\text{db}_{1}\wedge\text{db}_{2,3}+2\,b_{2,3}\otimes\text{db}_{2}\wedge\text{db}_{2,3}+2\,b_{1,3}\otimes\text{da}_{1,2}\wedge\text{da}_{2,3}+2\,b_{1,3}\otimes\text{db}_{1}\wedge\text{db}_{1,3}+\\\
&2\,b_{2,1}\otimes\text{da}_{2,2}\wedge\text{da}_{2,1}+2\,b_{2,1}\otimes\text{da}_{3,3}\wedge\text{da}_{2,1}+2\,b_{2,1}\otimes\text{da}_{2,3}\wedge\text{da}_{3,1}+b_{2,1}\otimes\text{db}_{1}\wedge\text{db}_{2,1}+2\,b_{2,1}\otimes\text{db}_{2}\wedge\text{db}_{2,1}\\\
&+b_{2,1}\otimes\text{db}_{2,3}\wedge\text{db}_{3,1}+b_{3,2}\otimes\text{da}_{2,2}\wedge\text{da}_{3,2}+b_{3,2}\otimes\text{da}_{3,3}\wedge\text{da}_{3,2}+b_{3,2}\otimes\text{da}_{1,2}\wedge\text{da}_{3,1}+b_{3,2}\otimes\text{db}_{1,2}\wedge\text{db}_{3,1}+\\\
&2\,b_{3,1}\otimes\text{da}_{2,1}\wedge\text{da}_{3,2}\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{psq}}(4)$, we have
$H^{2}({\mathfrak{g}};{\mathfrak{g}})=0$ for $p=3$ and for $p=5$.

We did not insert here the (interesting) result of deformations of
${\mathfrak{psq}}(2)$ because it is not simple.

##  4\. Results for $p=2$

If the shapes of the answers for the usual cohomology and divided power one do
not differ, we only give the answer for the usual one.

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{o}}^{(1)}(3)$, the space
$H^{2}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycles
$(\ref{eqo3})$:

(29) $\footnotesize\begin{array}[]{ll}\deg=-2:&y\otimes dh\wedge
dx\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{o}}{\mathfrak{o}}_{I\Pi}^{(1)}(1|2)$, the space
$H^{2}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycles
$(\ref{eqoo12})$:

(30) $\footnotesize\begin{array}[]{ll}\deg=-2:&h_{1}\otimes
d\left(x_{1}\right){}^{\wedge 2}+x_{1}\otimes\left(dx_{1}\wedge
dx_{2}\right)+y_{2}\otimes\left(dx_{1}\wedge
dy_{1}\right)+y_{2}\otimes\left(dx_{2}\wedge dy_{2}\right)\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{sl}}(3)$, we have
$H^{2}({\mathfrak{g}};{\mathfrak{g}})=0$ for any $p$.

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{o}}^{(1)}_{I}(5)$, we take the Cartan matrix

(31) $\begin{pmatrix}\overline{1}&1\\\ 1&{\bar{0}}\\\
\end{pmatrix}\qquad\begin{array}[]{l}\text{and the basis }\\\
x_{1},x_{2},x_{3}=[x_{1}\text{,$\,$}x_{2}],x_{4}=[x_{1}\text{,$\,$}[x_{1}\text{,$\,$}x_{2}]].\end{array}$

Then $H^{2}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycles
$(\ref{eqo5})$.

(32) $\tiny\begin{array}[]{ll}\deg=-4:&h_{1}\otimes dx_{2}\wedge
dx_{4}+x_{1}\otimes dx_{3}\wedge dx_{4}+y_{1}\otimes dx_{2}\wedge dx_{3}+\\\
&y_{2}\otimes dh_{2}\wedge dx_{4}+y_{3}\otimes dh_{2}\wedge
dx_{3}+y_{4}\otimes dh_{2}\wedge dx_{2}\\\ &\\\ \deg=-2:&h_{1}\otimes
dx_{4}\wedge dy_{2}+x_{2}\otimes dh_{1}\wedge dx_{4}+x_{2}\otimes dh_{2}\wedge
dx_{4}+x_{3}\otimes dx_{1}\wedge dx_{4}+y_{1}\otimes dh_{1}\wedge dx_{1}+\\\
&y_{1}\otimes dh_{2}\wedge dx_{1}+y_{1}\otimes dx_{3}\wedge
dy_{2}+y_{4}\otimes dh_{1}\wedge dy_{2}+y_{4}\otimes dh_{2}\wedge
dy_{2}+y_{4}\otimes dx_{1}\wedge dy_{3}\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{wk}}(3;\alpha)$, where $\alpha\neq 0,-1$, we
take the Cartan matrix

(33) $\begin{pmatrix}{\bar{0}}&\alpha&0\\\ \alpha&{\bar{0}}&1\\\
0&1&{\bar{0}}\end{pmatrix}\qquad\tiny\begin{array}[]{l}\text{and the basis
}\\\ x_{1},\;\;x_{2},\;\;x_{3},\\\ x_{4}=[x_{1}\text{,$\,$}x_{2}],\;\
x_{5}=[x_{2}\text{,$\,$}x_{3}],\\\
x_{6}=[x_{3}\text{,$\,$}[x_{1}\text{,$\,$}x_{2}]],\\\
x_{7}=[[x_{1}\text{,$\,$}x_{2}]\text{,$\,$}[x_{2}\text{,$\,$}x_{3}]].\end{array}$

Then $H^{2}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycles
$(\ref{eqwk3a})$.

(34) $\tiny\begin{array}[]{ll}\deg=-6:&\alpha(1+\alpha)y_{1}\otimes
dx_{3}\wedge dx_{7}+\alpha\,y_{1}\otimes dx_{5}\wedge
dx_{6}+\alpha(1+\alpha)y_{3}\otimes dx_{1}\wedge
dx_{7}+\alpha^{2}\,y_{3}\otimes dx_{4}\wedge dx_{6}+\\\ &\alpha\,y_{4}\otimes
dx_{3}\wedge dx_{6}+\alpha\,y_{5}\otimes dx_{1}\wedge dx_{6}+y_{6}\otimes
dx_{1}\wedge dx_{5}+\alpha\,y_{6}\otimes dx_{3}\wedge dx_{4}+y_{7}\otimes
dx_{1}\wedge dx_{3}\\\ &\\\ \deg=-4:&\alpha(1+\alpha)x_{1}\otimes dx_{3}\wedge
dx_{7}+\alpha\,x_{1}\otimes dx_{5}\wedge dx_{6}+\alpha\,y_{2}\otimes
dx_{3}\wedge dx_{5}+\alpha\,y_{3}\otimes dx_{2}\wedge
dx_{5}+\alpha(1+\alpha)y_{3}\otimes dx_{7}\wedge dy_{1}+\\\
&\alpha\,y_{5}\otimes dx_{2}\wedge dx_{3}+\alpha\,y_{5}\otimes dx_{6}\wedge
dy_{1}+y_{6}\otimes dx_{5}\wedge dy_{1}+y_{7}\otimes dx_{3}\wedge dy_{1}\\\
&\\\ \deg=-4:&\alpha(1+\alpha)x_{3}\otimes dx_{1}\wedge
dx_{7}+\alpha^{2}\,x_{3}\otimes dx_{4}\wedge dx_{6}+\alpha\,y_{1}\otimes
dx_{2}\wedge dx_{4}+\alpha(1+\alpha)y_{1}\otimes dx_{7}\wedge
dy_{3}+\alpha\,y_{2}\otimes dx_{1}\wedge dx_{4}+\\\ &y_{4}\otimes dx_{1}\wedge
dx_{2}+\alpha\,y_{4}\otimes dx_{6}\wedge dy_{3}+\alpha\,y_{6}\otimes
dx_{4}\wedge dy_{3}+y_{7}\otimes dx_{1}\wedge dy_{3}\\\ &\\\
\deg=-2:&\alpha\,x_{1}\otimes dx_{2}\wedge dx_{4}+\alpha(1+\alpha)x_{1}\otimes
dx_{7}\wedge dy_{3}+\alpha\,x_{3}\otimes dx_{2}\wedge
dx_{5}+\alpha(1+\alpha)x_{3}\otimes dx_{7}\wedge dy_{1}+\alpha\,y_{2}\otimes
dx_{4}\wedge dy_{1}+\\\ &\alpha\,y_{2}\otimes dx_{5}\wedge dy_{3}+y_{4}\otimes
dx_{2}\wedge dy_{1}+\alpha\,y_{5}\otimes dx_{2}\wedge dy_{3}+y_{7}\otimes
dy_{1}\wedge dy_{3}\\\ &\\\ \deg=0:&h_{1}\otimes\left(dx_{4}\wedge
dy_{4}\right)+h_{1}\otimes\left(dx_{6}\wedge
dy_{6}\right)+h_{1}\otimes\left(dx_{7}\wedge
dy_{7}\right)+\alpha\,h_{3}\otimes\left(dx_{7}\wedge
dy_{7}\right)+h_{4}\otimes\left(dx_{2}\wedge dy_{2}\right)\\\
&+\alpha\,h_{4}\otimes\left(dx_{4}\wedge
dy_{4}\right)+h_{4}\otimes\left(dx_{5}\wedge
dy_{5}\right)+\alpha\,h_{4}\otimes\left(dx_{6}\wedge
dy_{6}\right)+x_{1}\otimes\left(dx_{4}\wedge
dy_{2}\right)+x_{1}\otimes\left(dx_{6}\wedge dy_{5}\right)\\\
&+\alpha\,x_{2}\otimes\left(dx_{7}\wedge
dy_{6}\right)+x_{4}\otimes\left(dx_{7}\wedge
dy_{5}\right)+\alpha\,x_{5}\otimes\left(dx_{7}\wedge
dy_{4}\right)+x_{6}\otimes\left(dx_{7}\wedge
dy_{2}\right)+y_{1}\otimes\left(dx_{2}\wedge dy_{4}\right)\\\
&+y_{1}\otimes\left(dx_{5}\wedge
dy_{6}\right)+\alpha\,y_{2}\otimes\left(dx_{6}\wedge
dy_{7}\right)+y_{4}\otimes\left(dx_{5}\wedge
dy_{7}\right)+\alpha\,y_{5}\otimes\left(dx_{4}\wedge
dy_{7}\right)+y_{6}\otimes\left(dx_{2}\wedge dy_{7}\right)\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{bgl}}(3;\alpha)$, where $\alpha\neq 0,-1$, (the
super analog of ${\mathfrak{wk}}(3;\alpha)$ and an analog of non-existing
${\mathfrak{osp}}(4|2;\alpha)$ for $p=2$), we take the Cartan matrix

(35) $\begin{pmatrix}0&\alpha&0\\\ \alpha&{\bar{0}}&1\\\
0&1&{\bar{0}}\end{pmatrix}\qquad\tiny\begin{array}[]{l}\text{and the basis:
even $\mid$ odd}\\\ x_{2},\;\;x_{3}\mid x_{1},\\\
x_{5}=[x_{2}\text{,$\,$}x_{3}]\mid x_{4}=[x_{1}\text{,$\,$}x_{2}],\\\ \mid
x_{6}=[x_{3}\text{,$\,$}[x_{1}\text{,$\,$}x_{2}]],\\\ \mid
x_{7}=[[x_{1}\text{,$\,$}x_{2}]\text{,$\,$}[x_{2}\text{,$\,$}x_{3}]]\end{array}$

Then $H^{2}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycles
$(\ref{eqosp42a})$.

(36) $\tiny\begin{array}[]{ll}\deg=-8:&h_{1}\otimes dx_{7}^{\wedge
2}+\alpha\,h_{3}\otimes dx_{7}^{\wedge 2}\\\ &\\\ \deg=-6:&h_{1}\otimes
dx_{6}^{\wedge 2}+\alpha\,h_{3}\otimes dx_{6}^{\wedge 2}\\\ &\\\
\deg=-4:&\alpha(1+\alpha)x_{1}\otimes dx_{3}\wedge dx_{7}+\alpha\,x_{1}\otimes
dx_{5}\wedge dx_{6}+\alpha\,y_{2}\otimes dx_{3}\wedge
dx_{5}+\alpha\,y_{3}\otimes dx_{2}\wedge dx_{5}+\\\
&\alpha(1+\alpha)y_{3}\otimes dx_{7}\wedge dy_{1}+\alpha\,y_{5}\otimes
dx_{2}\wedge dx_{3}+\alpha\,y_{5}\otimes dx_{6}\wedge dy_{1}+y_{6}\otimes
dx_{5}\wedge dy_{1}+y_{7}\otimes dx_{3}\wedge dy_{1}\\\ &\\\
\deg=-4:&h_{1}\otimes dx_{4}^{\wedge 2}+\alpha\,h_{3}\otimes dx_{4}^{\wedge
2}\\\ &\\\ \deg=-4:&\alpha^{2}\,h_{2}\otimes dx_{4}{}^{\wedge
2}+\alpha^{2}\,h_{3}\otimes dx_{4}^{\wedge
2}+(\alpha^{3}+\alpha^{2})\,h_{4}\otimes dx_{4}^{\wedge
2}+(\alpha^{2}+\alpha)\,x_{3}\otimes\left(dx_{1}\wedge
dx_{7}\right)+\alpha^{2}\,x_{3}\otimes\left(dx_{4}\wedge dx_{6}\right)\\\
&+\alpha\,y_{1}\otimes\left(dx_{2}\wedge
dx_{4}\right)+(\alpha^{2}+\alpha)\,y_{1}\otimes\left(dx_{7}\wedge
dy_{3}\right)+\alpha\,y_{2}\otimes\left(dx_{1}\wedge
dx_{4}\right)+y_{4}\otimes\left(dx_{1}\wedge dx_{2}\right)\\\
&+\alpha\,y_{4}\otimes\left(dx_{6}\wedge
dy_{3}\right)+\alpha\,y_{6}\otimes\left(dx_{4}\wedge
dy_{3}\right)+y_{7}\otimes\left(dx_{1}\wedge dy_{3}\right)\\\ &\\\
\deg=-2:&h_{1}\otimes dx_{1}^{\wedge 2}+\alpha\,h_{3}\otimes dx_{1}^{\wedge
2}\\\ &\\\ \deg=-2:&\alpha\,x_{1}\otimes dx_{2}\wedge
dx_{4}+\alpha(1+\alpha)x_{1}\otimes dx_{7}\wedge dy_{3}+\alpha\,x_{3}\otimes
dx_{2}\wedge dx_{5}+\alpha(1+\alpha)x_{3}\otimes dx_{7}\wedge dy_{1}+\\\
&\alpha\,y_{2}\otimes dx_{4}\wedge dy_{1}+\alpha\,y_{2}\otimes dx_{5}\wedge
dy_{3}+y_{4}\otimes dx_{2}\wedge dy_{1}+\alpha\,y_{5}\otimes dx_{2}\wedge
dy_{3}+y_{7}\otimes dy_{1}\wedge dy_{3}\\\ &\\\
\deg=0:&h_{1}\otimes\left(dx_{4}\wedge
dy_{4}\right)+h_{1}\otimes\left(dx_{6}\wedge
dy_{6}\right)+h_{1}\otimes\left(dx_{7}\wedge
dy_{7}\right)+\alpha\,h_{3}\otimes\left(dx_{7}\wedge
dy_{7}\right)+h_{4}\otimes\left(dx_{2}\wedge dy_{2}\right)\\\
&+\alpha\,h_{4}\otimes\left(dx_{4}\wedge
dy_{4}\right)+h_{4}\otimes\left(dx_{5}\wedge
dy_{5}\right)+\alpha\,h_{4}\otimes\left(dx_{6}\wedge
dy_{6}\right)+x_{1}\otimes\left(dx_{4}\wedge
dy_{2}\right)+x_{1}\otimes\left(dx_{6}\wedge dy_{5}\right)\\\
&+\alpha\,x_{2}\otimes\left(dx_{7}\wedge
dy_{6}\right)+x_{4}\otimes\left(dx_{7}\wedge
dy_{5}\right)+\alpha\,x_{5}\otimes\left(dx_{7}\wedge
dy_{4}\right)+x_{6}\otimes\left(dx_{7}\wedge
dy_{2}\right)+y_{1}\otimes\left(dx_{2}\wedge dy_{4}\right)\\\
&+y_{1}\otimes\left(dx_{5}\wedge
dy_{6}\right)+\alpha\,y_{2}\otimes\left(dx_{6}\wedge
dy_{7}\right)+y_{4}\otimes\left(dx_{5}\wedge
dy_{7}\right)+\alpha\,y_{5}\otimes\left(dx_{4}\wedge
dy_{7}\right)+y_{6}\otimes\left(dx_{2}\wedge dy_{7}\right)\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{wk}}(4;\alpha)$, where $\alpha\neq 0,-1$, we
take the Cartan matrix and the basis

(37) $\begin{pmatrix}{\bar{0}}&\alpha&1&0\\\ \alpha&{\bar{0}}&0&0\\\
1&0&{\bar{0}}&1\\\
0&0&1&{\bar{0}}\end{pmatrix}\qquad\tiny\begin{array}[]{l}x_{1},\;\;x_{2},\;\;x_{3},\;\;x_{4},\\\
{}[x_{1}\text{,$\,$}x_{2}],\;\;[x_{1}\text{,$\,$}x_{3}],\;\;[x_{3}\text{,$\,$}x_{4}],\\\
{}[x_{3}\text{,$\,$}[x_{1}\text{,$\,$}x_{2}]],\;\;{}[x_{4}\text{,$\,$}[x_{1}\text{,$\,$}x_{3}]],\\\
{}[[x_{1}\text{,$\,$}x_{2}]\text{,$\,$}[x_{1}\text{,$\,$}x_{3}]],\;\;{}[[x_{1}\text{,$\,$}x_{2}]\text{,$\,$}[x_{3}\text{,$\,$}x_{4}]],\\\
{}[[x_{1}\text{,$\,$}x_{2}]\text{,$\,$}[x_{4}\text{,$\,$}[x_{1}\text{,$\,$}x_{3}]]],\\\
{}[[x_{3}\text{,$\,$}[x_{1}\text{,$\,$}x_{2}]]\text{,$\,$}[x_{4}\text{,$\,$}[x_{1}\text{,$\,$}x_{3}]]],\\\
{}[[x_{4}\text{,$\,$}[x_{1}\text{,$\,$}x_{3}]]\text{,$\,$}[[x_{1}\text{,$\,$}x_{2}]\text{,$\,$}[x_{1}\text{,$\,$}x_{3}]]],\\\
{}[[[x_{1}\text{,$\,$}x_{2}]\text{,$\,$}[x_{1}\text{,$\,$}x_{3}]]\text{,$\,$}[[x_{1}\text{,$\,$}x_{2}]\text{,$\,$}[x_{3}\text{,$\,$}x_{4}]]]\end{array}$

Then $H^{2}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycles which
for polygraphical reasons are divided into three groups: $(\ref{eqwk4a})$,
$(\ref{eqwk4ac1})$, $(\ref{eqwk4ac2})$, $(\ref{eqpsl4c3})$.

(38) $\tiny\begin{array}[]{ll}\deg=-12&\alpha^{3}(1+\alpha)y_{3}\otimes
dx_{9}\wedge dx_{15}+\alpha^{3}(1+\alpha)y_{3}\otimes dx_{11}\wedge
dx_{14}+\alpha^{2}(1+\alpha)y_{3}\otimes dx_{12}\wedge dx_{13}+\\\
&\alpha^{3}(1+\alpha)y_{6}\otimes dx_{7}\wedge
dx_{15}+\alpha^{2}(1+\alpha)y_{6}\otimes dx_{11}\wedge
dx_{13}+\alpha^{3}(1+\alpha)y_{7}\otimes dx_{6}\wedge dx_{15}+\\\
&\alpha^{3}(1+\alpha)y_{7}\otimes dx_{8}\wedge
dx_{14}+\alpha^{2}(1+\alpha)y_{7}\otimes dx_{10}\wedge
dx_{13}+\alpha^{2}(1+\alpha)y_{8}\otimes dx_{7}\wedge dx_{14}+\\\
&\alpha(1+\alpha)y_{8}\otimes dx_{9}\wedge
dx_{13}+\alpha^{3}(1+\alpha)y_{9}\otimes dx_{3}\wedge
dx_{15}+\alpha^{2}(1+\alpha)y_{9}\otimes dx_{8}\wedge dx_{13}+\\\
&\alpha(1+\alpha)y_{10}\otimes dx_{7}\wedge
dx_{13}+\alpha^{2}(1+\alpha)y_{11}\otimes dx_{3}\wedge
dx_{14}+\alpha(1+\alpha)y_{11}\otimes dx_{6}\wedge dx_{13}+\\\
&\alpha(1+\alpha)y_{12}\otimes dx_{3}\wedge
dx_{13}+\alpha(1+\alpha)y_{13}\otimes dx_{3}\wedge
dx_{12}+\alpha\,y_{13}\otimes dx_{6}\wedge dx_{11}+\\\
&\alpha(1+\alpha)y_{13}\otimes dx_{7}\wedge dx_{10}+\alpha\,y_{13}\otimes
dx_{8}\wedge dx_{9}+\alpha\,y_{14}\otimes dx_{3}\wedge dx_{11}+\\\
&\alpha\,y_{14}\otimes dx_{7}\wedge dx_{8}+y_{15}\otimes dx_{3}\wedge
dx_{9}+y_{15}\otimes dx_{6}\wedge dx_{7}\\\ &\\\
\deg=-10&\alpha^{3}(1+\alpha)x_{3}\otimes dx_{9}\wedge
dx_{15}+\alpha^{3}(1+\alpha)x_{3}\otimes dx_{11}\wedge
dx_{14}+\alpha^{2}(1+\alpha)x_{3}\otimes dx_{12}\wedge dx_{13}+\\\
&\alpha^{3}(1+\alpha)y_{1}\otimes dx_{4}\wedge
dx_{15}+\alpha^{2}(1+\alpha)y_{1}\otimes dx_{11}\wedge
dx_{12}+\alpha^{3}(1+\alpha)y_{4}\otimes dx_{1}\wedge dx_{15}+\\\
&\alpha^{3}(1+\alpha)y_{4}\otimes dx_{5}\wedge
dx_{14}+\alpha^{2}(1+\alpha)y_{4}\otimes dx_{10}\wedge
dx_{12}+\alpha^{2}(1+\alpha)y_{5}\otimes dx_{4}\wedge dx_{14}+\\\
&\alpha(1+\alpha)y_{5}\otimes dx_{9}\wedge
dx_{12}+\alpha^{2}(1+\alpha)y_{9}\otimes dx_{5}\wedge
dx_{12}+\alpha^{3}(1+\alpha)y_{9}\otimes dx_{15}\wedge dy_{3}+\\\
&\alpha(1+\alpha)y_{10}\otimes dx_{4}\wedge
dx_{12}+\alpha(1+\alpha)y_{11}\otimes dx_{1}\wedge dx_{12}+\\\
&\alpha^{2}(1+\alpha)y_{11}\otimes dx_{14}\wedge dy_{3}+\alpha\,y_{12}\otimes
dx_{1}\wedge dx_{11}+\alpha(1+\alpha)y_{12}\otimes dx_{4}\wedge dx_{10}+\\\
&\alpha\,y_{12}\otimes dx_{5}\wedge dx_{9}+\alpha(1+\alpha)y_{12}\otimes
dx_{13}\wedge dy_{3}+\alpha(1+\alpha)y_{13}\otimes dx_{12}\wedge dy_{3}+\\\
&\alpha\,y_{14}\otimes dx_{4}\wedge dx_{5}+\alpha\,y_{14}\otimes dx_{11}\wedge
dy_{3}+y_{15}\otimes dx_{1}\wedge dx_{4}+y_{15}\otimes dx_{9}\wedge dy_{3}\\\
&\\\ \deg=-8&\alpha^{3}(1+\alpha)x_{4}\otimes dx_{1}\wedge
dx_{15}+\alpha^{3}(1+\alpha)x_{4}\otimes dx_{5}\wedge
dx_{14}+\alpha^{2}(1+\alpha)x_{4}\otimes dx_{10}\wedge dx_{12}+\\\
&\alpha^{3}(1+\alpha)x_{7}\otimes dx_{6}\wedge
dx_{15}+\alpha^{3}(1+\alpha)x_{7}\otimes dx_{8}\wedge
dx_{14}+\alpha^{3}(1+\alpha)x_{7}\otimes dx_{10}\wedge dx_{13}+\\\
&\alpha^{2}(1+\alpha)y_{1}\otimes dx_{8}\wedge
dx_{10}+\alpha^{3}(1+\alpha)y_{1}\otimes dx_{15}\wedge
dy_{4}+\alpha(1+\alpha)y_{5}\otimes dx_{6}\wedge dx_{10}+\\\
&\alpha^{2}(1+\alpha)y_{5}\otimes dx_{14}\wedge
dy_{4}+\alpha^{2}(1+\alpha)y_{6}\otimes dx_{5}\wedge
dx_{10}+\alpha^{3}(1+\alpha)y_{6}\otimes dx_{15}\wedge dy_{7}+\\\
&\alpha(1+\alpha)y_{8}\otimes dx_{1}\wedge
dx_{10}+\alpha^{2}(1+\alpha)y_{8}\otimes dx_{14}\wedge
dy_{7}+\alpha\,y_{10}\otimes dx_{1}\wedge dx_{8}+\\\ &\alpha\,y_{10}\otimes
dx_{5}\wedge dx_{6}+\alpha(1+\alpha)y_{10}\otimes dx_{12}\wedge
dy_{4}+\alpha(1+\alpha)\,y_{10}\otimes dx_{13}\wedge dy_{7}+\\\
&\alpha(1+\alpha)y_{12}\otimes dx_{10}\wedge
dy_{4}+\alpha(1+\alpha)y_{13}\otimes dx_{10}\wedge
dy_{7}+\alpha\,y_{14}\otimes dx_{5}\wedge dy_{4}+\\\ &\alpha\,y_{14}\otimes
dx_{8}\wedge dy_{7}+y_{15}\otimes dx_{1}\wedge dy_{4}+y_{15}\otimes
dx_{6}\wedge dy_{7}\\\ &\\\ \end{array}$ (39)
$\tiny\begin{array}[]{ll}\deg=-8&\alpha^{3}(1+\alpha)x_{1}\otimes dx_{4}\wedge
dx_{15}+\alpha^{2}(1+\alpha)x_{1}\otimes dx_{11}\wedge
dx_{12}+\alpha^{3}(1+\alpha)x_{6}\otimes dx_{7}\wedge dx_{15}+\\\
&\alpha^{2}(1+\alpha)x_{6}\otimes dx_{11}\wedge
dx_{13}+\alpha^{2}(1+\alpha)y_{2}\otimes dx_{4}\wedge
dx_{13}+\alpha^{2}(1+\alpha)y_{2}\otimes dx_{7}\wedge dx_{12}+\\\
&\alpha^{2}(1+\alpha)y_{4}\otimes dx_{2}\wedge
dx_{13}+\alpha^{2}(1+\alpha)y_{4}\otimes dx_{8}\wedge
dx_{11}+\alpha^{3}(1+\alpha)y_{4}\otimes dx_{15}\wedge dy_{1}+\\\
&\alpha(1+\alpha)y_{5}\otimes dx_{7}\wedge
dx_{11}+\alpha^{2}(1+\alpha)y_{7}\otimes dx_{2}\wedge
dx_{12}+\alpha^{2}(1+\alpha)y_{7}\otimes dx_{5}\wedge dx_{11}+\\\
&\alpha^{3}(1+\alpha)y_{7}\otimes dx_{15}\wedge
dy_{6}+\alpha(1+\alpha)y_{8}\otimes dx_{4}\wedge
dx_{11}+\alpha(1+\alpha)y_{11}\otimes dx_{4}\wedge dx_{8}+\\\
&\alpha(1+\alpha)y_{11}\otimes dx_{5}\wedge
dx_{7}+\alpha(1+\alpha)y_{11}\otimes dx_{12}\wedge
dy_{1}+\alpha(1+\alpha)\,y_{11}\otimes dx_{13}\wedge dy_{6}+\\\
&\alpha\,y_{12}\otimes dx_{2}\wedge dx_{7}+\alpha\,y_{12}\otimes dx_{11}\wedge
dy_{1}+\alpha\,y_{13}\otimes dx_{2}\wedge dx_{4}+\\\ &\alpha\,y_{13}\otimes
dx_{11}\wedge dy_{6}+y_{15}\otimes dx_{4}\wedge dy_{1}+y_{15}\otimes
dx_{7}\wedge dy_{6}\\\ &\\\ \deg=-6&\alpha(1+\alpha)x_{2}\otimes dx_{4}\wedge
dx_{13}+\alpha(1+\alpha)x_{2}\otimes dx_{7}\wedge
dx_{12}+\alpha(1+\alpha)x_{5}\otimes dx_{4}\wedge dx_{14}+\\\
&(\alpha+1)\,x_{5}\otimes dx_{9}\wedge dx_{12}+\alpha(1+\alpha)x_{8}\otimes
dx_{7}\wedge dx_{14}+(\alpha+1)\,x_{8}\otimes dx_{9}\wedge dx_{13}+\\\
&(\alpha+1)\,y_{1}\otimes dx_{7}\wedge dx_{9}+(\alpha+1)\,y_{4}\otimes
dx_{6}\wedge dx_{9}+\alpha(1+\alpha)y_{4}\otimes dx_{13}\wedge dy_{2}+\\\
&\alpha^{2}(1+\alpha)y_{4}\otimes dx_{14}\wedge
dy_{5}+(\alpha+1)\,y_{6}\otimes dx_{4}\wedge dx_{9}+(\alpha+1)\,y_{7}\otimes
dx_{1}\wedge dx_{9}+\\\ &\alpha(1+\alpha)y_{7}\otimes dx_{12}\wedge
dy_{2}+\alpha^{2}(1+\alpha)y_{7}\otimes dx_{14}\wedge
dy_{8}+(\alpha+1)\,y_{9}\otimes dx_{1}\wedge dx_{7}+\\\
&(\alpha+1)\,y_{9}\otimes dx_{4}\wedge dx_{6}+\alpha(1+\alpha)y_{9}\otimes
dx_{12}\wedge dy_{5}+\alpha(1+\alpha)y_{9}\otimes dx_{13}\wedge dy_{8}+\\\
&y_{12}\otimes dx_{7}\wedge dy_{2}+y_{12}\otimes dx_{9}\wedge
dy_{5}+y_{13}\otimes dx_{4}\wedge dy_{2}+y_{13}\otimes dx_{9}\wedge dy_{8}+\\\
&y_{14}\otimes dx_{4}\wedge dy_{5}+y_{14}\otimes dx_{7}\wedge dy_{8}\\\ &\\\
\deg=-6&\alpha^{2}(1+\alpha)x_{1}\otimes dx_{8}\wedge
dx_{10}+\alpha^{3}(1+\alpha)x_{1}\otimes dx_{15}\wedge
dy_{4}+\alpha^{2}(1+\alpha)x_{4}\otimes dx_{2}\wedge dx_{13}+\\\
&\alpha^{2}(1+\alpha)x_{4}\otimes dx_{8}\wedge
dx_{11}+\alpha^{3}(1+\alpha)x_{4}\otimes dx_{15}\wedge
dy_{1}+\alpha^{3}(1+\alpha)x_{9}\otimes dx_{3}\wedge dx_{15}+\\\
&\alpha^{2}(1+\alpha)x_{9}\otimes dx_{8}\wedge
dx_{13}+\alpha^{2}(1+\alpha)y_{2}\otimes dx_{3}\wedge
dx_{10}+\alpha^{2}(1+\alpha)y_{2}\otimes dx_{13}\wedge dy_{4}+\\\
&\alpha^{2}(1+\alpha)y_{3}\otimes dx_{2}\wedge
dx_{10}+\alpha^{2}(1+\alpha)y_{3}\otimes dx_{5}\wedge
dx_{8}+\alpha^{3}(1+\alpha)y_{3}\otimes dx_{15}\wedge dy_{9}+\\\
&\alpha(1+\alpha)y_{5}\otimes dx_{3}\wedge dx_{8}+\alpha(1+\alpha)y_{8}\otimes
dx_{3}\wedge dx_{5}+\alpha(1+\alpha)y_{8}\otimes dx_{10}\wedge dy_{1}+\\\
&\alpha(1+\alpha)y_{8}\otimes dx_{11}\wedge
dy_{4}+\alpha(1+\alpha)y_{8}\otimes dx_{13}\wedge dy_{9}+\alpha\,y_{10}\otimes
dx_{2}\wedge dx_{3}+\\\ &\alpha\,y_{10}\otimes dx_{8}\wedge
dy_{1}+\alpha(1+\alpha)y_{11}\otimes dx_{8}\wedge dy_{4}+\alpha\,y_{13}\otimes
dx_{2}\wedge dy_{4}+\\\ &\alpha\,y_{13}\otimes dx_{8}\wedge
dy_{9}+y_{15}\otimes dx_{3}\wedge dy_{9}+y_{15}\otimes dy_{1}\wedge dy_{4}\\\
\end{array}$

(40) $\tiny\begin{array}[]{ll}\deg=0&\alpha\,h_{1}\otimes dx_{14}\wedge
dy_{14}+\alpha^{2}\,h_{1}\otimes dx_{15}\wedge dy_{15}+h_{2}\otimes
dx_{5}\wedge dy_{5}+h_{2}\otimes dx_{8}\wedge dy_{8}+\\\ &h_{2}\otimes
dx_{10}\wedge dy_{10}+h_{2}\otimes dx_{11}\wedge dy_{11}+h_{2}\otimes
dx_{12}\wedge dy_{12}+h_{2}\otimes dx_{13}\wedge dy_{13}+\\\
&\alpha^{2}\,h_{2}\otimes dx_{14}\wedge dy_{14}+h_{3}\otimes dx_{1}\wedge
dy_{1}+\alpha\,h_{3}\otimes dx_{5}\wedge dy_{5}+h_{3}\otimes dx_{6}\wedge
dy_{6}+\\\ &\alpha\,h_{3}\otimes dx_{8}\wedge dy_{8}+h_{3}\otimes dx_{9}\wedge
dy_{9}+\alpha\,h_{3}\otimes dx_{10}\wedge dy_{10}+\alpha\,h_{3}\otimes
dx_{11}\wedge dy_{11}+\\\ &\alpha\,h_{3}\otimes dx_{12}\wedge
dy_{12}+\alpha^{2}(1+\alpha)h_{3}\otimes dx_{14}\wedge
dy_{14}+\alpha^{3}(1+\alpha)h_{3}\otimes dx_{15}\wedge dy_{15}+\\\
&\alpha\,h_{4}\otimes dx_{12}\wedge dy_{12}+\alpha\,h_{4}\otimes dx_{13}\wedge
dy_{13}+\alpha\,h_{4}\otimes dx_{14}\wedge dy_{14}+\alpha^{2}\,h_{4}\otimes
dx_{15}\wedge dy_{15}+\\\ &x_{1}\otimes dh_{1}\wedge
dx_{1}+\alpha\,x_{1}\otimes dx_{10}\wedge dy_{8}+\alpha\,x_{1}\otimes
dx_{12}\wedge dy_{11}+\alpha\,x_{1}\otimes dx_{14}\wedge dy_{13}+\\\
&x_{2}\otimes dh_{1}\wedge dx_{2}+x_{2}\otimes dx_{5}\wedge
dy_{1}+x_{2}\otimes dx_{8}\wedge dy_{6}+x_{2}\otimes dx_{11}\wedge
dy_{9}+\alpha^{3}\,x_{2}\otimes dx_{15}\wedge dy_{14}+\\\
&\alpha\,x_{3}\otimes dx_{13}\wedge dy_{12}+x_{4}\otimes dh_{1}\wedge
dx_{4}+\alpha\,x_{4}\otimes dx_{12}\wedge dy_{10}+x_{5}\otimes dx_{10}\wedge
dy_{6}+\\\ &x_{5}\otimes dx_{12}\wedge dy_{9}+\alpha\,x_{5}\otimes
dx_{15}\wedge dy_{13}+x_{6}\otimes dh_{1}\wedge dx_{6}+\alpha\,x_{6}\otimes
dx_{10}\wedge dy_{5}+\\\ &\alpha\,x_{6}\otimes dx_{13}\wedge
dy_{11}+\alpha\,x_{6}\otimes dx_{14}\wedge dy_{12}+x_{7}\otimes dh_{1}\wedge
dx_{7}+\alpha\,x_{7}\otimes dx_{13}\wedge dy_{10}+\\\ &x_{8}\otimes
dx_{10}\wedge dy_{1}+x_{8}\otimes dx_{13}\wedge dy_{9}+\alpha\,x_{8}\otimes
dx_{15}\wedge dy_{12}+\\\ &\alpha\,x_{9}\otimes dx_{12}\wedge
dy_{5}+\alpha\,x_{9}\otimes dx_{13}\wedge dy_{8}+\alpha\,x_{9}\otimes
dx_{14}\wedge dy_{10}+\\\ &x_{10}\otimes dh_{1}\wedge dx_{10}+x_{10}\otimes
dx_{14}\wedge dy_{9}+\alpha\,x_{10}\otimes dx_{15}\wedge dy_{11}+\\\
&x_{11}\otimes dh_{1}\wedge dx_{11}+x_{11}\otimes dx_{12}\wedge
dy_{1}+x_{11}\otimes dx_{13}\wedge dy_{6}+\alpha\,x_{11}\otimes dx_{15}\wedge
dy_{10}+\\\ &x_{12}\otimes dx_{14}\wedge dy_{6}+\alpha\,x_{12}\otimes
dx_{15}\wedge dy_{8}+x_{13}\otimes dx_{14}\wedge dy_{1}+\alpha\,x_{13}\otimes
dx_{15}\wedge dy_{5}+\\\ &x_{14}\otimes dh_{1}\wedge dx_{14}+y_{1}\otimes
dh_{1}\wedge dy_{1}+\alpha\,y_{1}\otimes dx_{8}\wedge
dy_{10}+\alpha\,y_{1}\otimes dx_{11}\wedge dy_{12}+\\\ &\alpha\,y_{1}\otimes
dx_{13}\wedge dy_{14}+y_{2}\otimes dh_{1}\wedge dy_{2}+y_{2}\otimes
dx_{1}\wedge dy_{5}+y_{2}\otimes dx_{6}\wedge dy_{8}+\\\ &y_{2}\otimes
dx_{9}\wedge dy_{11}+\alpha^{3}\,y_{2}\otimes dx_{14}\wedge
dy_{15}+\alpha\,y_{3}\otimes dx_{12}\wedge dy_{13}+y_{4}\otimes dh_{1}\wedge
dy_{4}\\\ &\alpha\,y_{4}\otimes dx_{10}\wedge dy_{12}+y_{5}\otimes
dx_{6}\wedge dy_{10}+y_{5}\otimes dx_{9}\wedge dy_{12}+\alpha\,y_{5}\otimes
dx_{13}\wedge dy_{15}+\\\ &y_{6}\otimes dh_{1}\wedge
dy_{6}+\alpha\,y_{6}\otimes dx_{5}\wedge dy_{10}+\alpha\,y_{6}\otimes
dx_{11}\wedge dy_{13}+\alpha\,y_{6}\otimes dx_{12}\wedge dy_{14}+\\\
&y_{7}\otimes dh_{1}\wedge dy_{7}+\alpha\,y_{7}\otimes dx_{10}\wedge
dy_{13}+y_{8}\otimes dx_{1}\wedge dy_{10}+y_{8}\otimes dx_{9}\wedge
dy_{13}+\\\ &\alpha\,y_{8}\otimes dx_{12}\wedge dy_{15}+\alpha\,y_{9}\otimes
dx_{5}\wedge dy_{12}+\alpha\,y_{9}\otimes dx_{8}\wedge
dy_{13}+\alpha\,y_{9}\otimes dx_{10}\wedge dy_{14}+\\\ &y_{10}\otimes
dh_{1}\wedge dy_{10}+y_{10}\otimes dx_{9}\wedge dy_{14}+\alpha\,y_{10}\otimes
dx_{11}\wedge dy_{15}+y_{11}\otimes dh_{1}\wedge dy_{11}+\\\ &y_{11}\otimes
dx_{1}\wedge dy_{12}+y_{11}\otimes dx_{6}\wedge dy_{13}+\alpha\,y_{11}\otimes
dx_{10}\wedge dy_{15}+\\\ &y_{12}\otimes dx_{6}\wedge
dy_{14}+\alpha\,y_{12}\otimes dx_{8}\wedge dy_{15}+y_{13}\otimes dx_{1}\wedge
dy_{14}+\\\ &\alpha\,y_{13}\otimes dx_{5}\wedge dy_{15}+y_{14}\otimes
dh_{1}\wedge dy_{14}\end{array}$ (41)
$\tiny\begin{array}[]{ll}\deg=-4&\alpha(1+\alpha)x_{2}\otimes dx_{3}\wedge
dx_{10}+\alpha(1+\alpha)x_{2}\otimes dx_{13}\wedge
dy_{4}+(\alpha+1)\,x_{4}\otimes dx_{6}\wedge dx_{9}+\\\
&\alpha(1+\alpha)x_{4}\otimes dx_{13}\wedge
dy_{2}+\alpha^{2}(1+\alpha)x_{4}\otimes dx_{14}\wedge
dy_{5}+(\alpha+1)\,x_{5}\otimes dx_{6}\wedge dx_{10}+\\\
&\alpha(1+\alpha)x_{5}\otimes dx_{14}\wedge
dy_{4}+\alpha(1+\alpha)x_{11}\otimes dx_{3}\wedge
dx_{14}+(\alpha+1)\,x_{11}\otimes dx_{6}\wedge dx_{13}+\\\
&(\alpha+1)\,y_{1}\otimes dx_{3}\wedge dx_{6}+(\alpha+1)\,y_{3}\otimes
dx_{1}\wedge dx_{6}+\alpha(1+\alpha)y_{3}\otimes dx_{10}\wedge dy_{2}+\\\
&\alpha^{2}(1+\alpha)y_{3}\otimes dx_{14}\wedge
dy_{11}+(\alpha+1)\,y_{6}\otimes dx_{1}\wedge dx_{3}+(\alpha+1)\,y_{6}\otimes
dx_{9}\wedge dy_{4}+\\\ &\alpha(1+\alpha)y_{6}\otimes dx_{10}\wedge
dy_{5}+\alpha(1+\alpha)y_{6}\otimes dx_{13}\wedge
dy_{11}+(\alpha+1)\,y_{9}\otimes dx_{6}\wedge dy_{4}+\\\ &y_{10}\otimes
dx_{3}\wedge dy_{2}+y_{10}\otimes dx_{6}\wedge dy_{5}+y_{13}\otimes
dx_{6}\wedge dy_{11}+\\\ &y_{13}\otimes dy_{2}\wedge dy_{4}+y_{14}\otimes
dx_{3}\wedge dy_{11}+y_{14}\otimes dy_{4}\wedge dy_{5}\\\ &\\\
\deg=-4&\alpha^{2}(1+\alpha)x_{3}\otimes dx_{2}\wedge
dx_{10}+\alpha^{2}(1+\alpha)x_{3}\otimes dx_{5}\wedge
dx_{8}+\alpha^{3}(1+\alpha)x_{3}\otimes dx_{15}\wedge dy_{9}+\\\
&\alpha^{2}(1+\alpha)x_{6}\otimes dx_{5}\wedge
dx_{10}+\alpha^{3}(1+\alpha)x_{6}\otimes dx_{15}\wedge
dy_{7}+\alpha^{2}(1+\alpha)x_{7}\otimes dx_{2}\wedge dx_{12}+\\\
&\alpha^{2}(1+\alpha)x_{7}\otimes dx_{5}\wedge
dx_{11}+\alpha^{3}(1+\alpha)x_{7}\otimes dx_{15}\wedge
dy_{6}+\alpha^{2}(1+\alpha)x_{9}\otimes dx_{5}\wedge dx_{12}+\\\
&\alpha^{3}(1+\alpha)x_{9}\otimes dx_{15}\wedge
dy_{3}+\alpha^{2}(1+\alpha)y_{2}\otimes dx_{10}\wedge
dy_{3}+\alpha^{2}(1+\alpha)y_{2}\otimes dx_{12}\wedge dy_{7}+\\\
&\alpha(1+\alpha)y_{5}\otimes dx_{8}\wedge dy_{3}+\alpha(1+\alpha)y_{5}\otimes
dx_{10}\wedge dy_{6}+\alpha(1+\alpha)y_{5}\otimes dx_{11}\wedge dy_{7}+\\\
&\alpha(1+\alpha)y_{5}\otimes dx_{12}\wedge
dy_{9}+\alpha(1+\alpha)y_{8}\otimes dx_{5}\wedge dy_{3}+\alpha\,y_{10}\otimes
dx_{2}\wedge dy_{3}+\\\ &\alpha\,y_{10}\otimes dx_{5}\wedge
dy_{6}+\alpha^{2}+\alpha\,y_{11}\otimes dx_{5}\wedge
dy_{7}+\alpha\,y_{12}\otimes dx_{2}\wedge dy_{7}+\\\ &\alpha\,y_{12}\otimes
dx_{5}\wedge dy_{9}+y_{15}\otimes dy_{3}\wedge dy_{9}+y_{15}\otimes
dy_{6}\wedge dy_{7}\\\ &\\\ \deg=-2&\alpha(1+\alpha)x_{2}\otimes dx_{10}\wedge
dy_{3}+\alpha(1+\alpha)x_{2}\otimes dx_{12}\wedge
dy_{7}+(\alpha+1)\,x_{3}\otimes dx_{1}\wedge dx_{6}+\\\
&\alpha^{2}+\alpha\,x_{3}\otimes dx_{10}\wedge
dy_{2}+\alpha^{3}+\alpha^{2}\,x_{3}\otimes dx_{14}\wedge
dy_{11}+(\alpha+1)\,x_{7}\otimes dx_{1}\wedge dx_{9}+\\\
&\alpha(1+\alpha)x_{7}\otimes dx_{12}\wedge
dy_{2}+\alpha^{2}(1+\alpha)x_{7}\otimes dx_{14}\wedge
dy_{8}+(\alpha+1)\,x_{8}\otimes dx_{1}\wedge dx_{10}+\\\
&\alpha(1+\alpha)x_{8}\otimes dx_{14}\wedge dy_{7}+(\alpha+1)\,x_{11}\otimes
dx_{1}\wedge dx_{12}+\alpha(1+\alpha)x_{11}\otimes dx_{14}\wedge dy_{3}+\\\
&(\alpha+1)\,y_{1}\otimes dx_{6}\wedge dy_{3}+(\alpha+1)\,y_{1}\otimes
dx_{9}\wedge dy_{7}+\alpha^{2}+\alpha\,y_{1}\otimes dx_{10}\wedge dy_{8}+\\\
&\alpha(1+\alpha)y_{1}\otimes dx_{12}\wedge dy_{11}+(\alpha+1)\,y_{6}\otimes
dx_{1}\wedge dy_{3}+(\alpha+1)\,y_{9}\otimes dx_{1}\wedge dy_{7}+\\\
&y_{10}\otimes dx_{1}\wedge dy_{8}+y_{10}\otimes dy_{2}\wedge
dy_{3}+y_{12}\otimes dx_{1}\wedge dy_{11}+y_{12}\otimes dy_{2}\wedge
dy_{7}+\\\ &y_{14}\otimes dy_{3}\wedge dy_{11}+y_{14}\otimes dy_{7}\wedge
dy_{8}\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{bgl}}(4;\alpha)$, where $\alpha\neq 0,-1$, we
take the Cartan matrix and the basis

(42) $\begin{pmatrix}0&\alpha&1&0\\\ \alpha&{\bar{0}}&0&0\\\
1&0&{\bar{0}}&1\\\
0&0&1&{\bar{0}}\end{pmatrix}\qquad\tiny\begin{array}[]{l}x_{1},\;\;x_{2},\;\;x_{3},\;\;x_{4},\\\
{}[x_{1}\text{,$\,$}x_{2}],\;\;[x_{1}\text{,$\,$}x_{3}],\;\;[x_{3}\text{,$\,$}x_{4}],\\\
{}[x_{3}\text{,$\,$}[x_{1}\text{,$\,$}x_{2}]],\;\;{}[x_{4}\text{,$\,$}[x_{1}\text{,$\,$}x_{3}]],\\\
{}[[x_{1}\text{,$\,$}x_{2}]\text{,$\,$}[x_{1}\text{,$\,$}x_{3}]],\;\;{}[[x_{1}\text{,$\,$}x_{2}]\text{,$\,$}[x_{3}\text{,$\,$}x_{4}]],\\\
{}[[x_{1}\text{,$\,$}x_{2}]\text{,$\,$}[x_{4}\text{,$\,$}[x_{1}\text{,$\,$}x_{3}]]],\\\
{}[[x_{3}\text{,$\,$}[x_{1}\text{,$\,$}x_{2}]]\text{,$\,$}[x_{4}\text{,$\,$}[x_{1}\text{,$\,$}x_{3}]]],\\\
{}[[x_{4}\text{,$\,$}[x_{1}\text{,$\,$}x_{3}]]\text{,$\,$}[[x_{1}\text{,$\,$}x_{2}]\text{,$\,$}[x_{1}\text{,$\,$}x_{3}]]],\\\
{}[[[x_{1}\text{,$\,$}x_{2}]\text{,$\,$}[x_{1}\text{,$\,$}x_{3}]]\text{,$\,$}[[x_{1}\text{,$\,$}x_{2}]\text{,$\,$}[x_{3}\text{,$\,$}x_{4}]]]\end{array}$

Then $H^{2}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycles:

(43)
$\tiny\begin{array}[]{ll}\deg=-12&\left(\alpha^{4}+\alpha^{3}\right)\,y_{3}\otimes\left(dx_{9}\wedge
dx_{15}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,y_{3}\otimes\left(dx_{11}\wedge
dx_{14}\right)\text{+
}\left(\alpha^{4}+\alpha^{2}\right)\,y_{3}\otimes\left(dx_{12}\wedge
dx_{13}\right)\\\ &\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,y_{6}\otimes\left(dx_{7}\wedge
dx_{15}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{6}\otimes\left(dx_{11}\wedge
dx_{13}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,y_{7}\otimes\left(dx_{6}\wedge
dx_{15}\right)\\\ &\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,y_{7}\otimes\left(dx_{8}\wedge
dx_{14}\right)\text{+
}\left(\alpha^{4}+\alpha^{2}\right)\,y_{7}\otimes\left(dx_{10}\wedge
dx_{13}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{8}\otimes\left(dx_{7}\wedge
dx_{14}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{8}\otimes\left(dx_{9}\wedge
dx_{13}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,y_{9}\otimes\left(dx_{3}\wedge
dx_{15}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{9}\otimes\left(dx_{8}\wedge
dx_{13}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{10}\otimes\left(dx_{7}\wedge
dx_{13}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{11}\otimes\left(dx_{3}\wedge
dx_{14}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{11}\otimes\left(dx_{6}\wedge
dx_{13}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{12}\otimes\left(dx_{3}\wedge
dx_{13}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{13}\otimes\left(dx_{3}\wedge
dx_{12}\right)\text{+ }\alpha\,y_{13}\otimes\left(dx_{6}\wedge
dx_{11}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{13}\otimes\left(dx_{7}\wedge
dx_{10}\right)\text{+ }\alpha\,y_{13}\otimes\left(dx_{8}\wedge
dx_{9}\right)\text{+ }\alpha\,y_{14}\otimes\left(dx_{3}\wedge
dx_{11}\right)\text{+ }\alpha\,y_{14}\otimes\left(dx_{7}\wedge
dx_{8}\right)\\\ &\text{+ }y_{15}\otimes\left(dx_{3}\wedge
dx_{9}\right)\text{+ }y_{15}\otimes\left(dx_{6}\wedge dx_{7}\right)\\\ &\\\
\deg=-10&\left(\alpha^{4}+\alpha^{3}\right)\,x_{3}\otimes\left(dx_{9}\wedge
dx_{15}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,x_{3}\otimes\left(dx_{11}\wedge
dx_{14}\right)\text{+
}\left(\alpha^{4}+\alpha^{2}\right)\,x_{3}\otimes\left(dx_{12}\wedge
dx_{13}\right)\\\ &\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,y_{1}\otimes\left(dx_{4}\wedge
dx_{15}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{1}\otimes\left(dx_{11}\wedge
dx_{12}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,y_{4}\otimes\left(dx_{1}\wedge
dx_{15}\right)\\\ &\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,y_{4}\otimes\left(dx_{5}\wedge
dx_{14}\right)\text{+
}\left(\alpha^{4}+\alpha^{2}\right)\,y_{4}\otimes\left(dx_{10}\wedge
dx_{12}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{5}\otimes\left(dx_{4}\wedge
dx_{14}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{5}\otimes\left(dx_{9}\wedge
dx_{12}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{9}\otimes\left(dx_{5}\wedge
dx_{12}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,y_{9}\otimes\left(dx_{15}\wedge
dy_{3}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{10}\otimes\left(dx_{4}\wedge
dx_{12}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{11}\otimes\left(dx_{1}\wedge
dx_{12}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{11}\otimes\left(dx_{14}\wedge
dy_{3}\right)\\\ &\text{+ }\alpha\,y_{12}\otimes\left(dx_{1}\wedge
dx_{11}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{12}\otimes\left(dx_{4}\wedge
dx_{10}\right)\text{+ }\alpha\,y_{12}\otimes\left(dx_{5}\wedge
dx_{9}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{12}\otimes\left(dx_{13}\wedge
dy_{3}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{13}\otimes\left(dx_{12}\wedge
dy_{3}\right)\text{+ }\alpha\,y_{14}\otimes\left(d_{4}\wedge
dx_{5}\right)\text{+ }\alpha\,y_{14}\otimes\left(dx_{11}\wedge
dy_{3}\right)\\\ &\text{+ }y_{15}\otimes\left(dx_{1}\wedge
dx_{4}\right)\text{+ }y_{15}\otimes\left(d_{9}\wedge dy_{3}\right)\\\ &\\\
\deg=-8&\left(\alpha^{4}+\alpha^{3}\right)\,x_{4}\otimes\left(dx_{1}\wedge
dx_{15}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,x_{4}\otimes\left(dx_{5}\wedge
dx_{14}\right)\text{+
}\left(\alpha^{4}+\alpha^{2}\right)\,x_{4}\otimes\left(dx_{10}\wedge
dx_{12}\right)\\\ &\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,x_{7}\otimes\left(dx_{6}\wedge
dx_{15}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,x_{7}\otimes\left(dx_{8}\wedge
dx_{14}\right)\text{+
}\left(\alpha^{4}+\alpha^{2}\right)\,x_{7}\otimes\left(dx_{10}\wedge
dx_{13}\right)\\\ &\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{1}\otimes\left(dx_{8}\wedge
dx_{10}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,y_{1}\otimes\left(dx_{15}\wedge
dy_{4}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{5}\otimes\left(dx_{6}\wedge
dx_{10}\right)\\\ &\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{5}\otimes\left(dx_{14}\wedge
dy_{4}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{6}\otimes\left(dx_{5}\wedge
dx_{10}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,y_{6}\otimes\left(dx_{15}\wedge
dy_{7}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{8}\otimes\left(dx_{1}\wedge
dx_{10}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{8}\otimes\left(dx_{14}\wedge
dy_{7}\right)\text{+ }\alpha\,y_{10}\otimes\left(dx_{1}\wedge dx_{8}\right)\\\
&\text{+ }\alpha\,y_{10}\otimes\left(dx_{5}\wedge dx_{6}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{10}\otimes\left(dx_{12}\wedge
dy_{4}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{10}\otimes\left(dx_{13}\wedge
dy_{7}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{12}\otimes\left(dx_{10}\wedge
dy_{4}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{13}\otimes\left(dx_{10}\wedge
dy_{7}\right)\text{+ }\alpha\,y_{14}\otimes\left(dx_{5}\wedge dy_{4}\right)\\\
&\text{+ }\alpha\,y_{14}\otimes\left(dx_{8}\wedge dy_{7}\right)\text{+
}y_{15}\otimes\left(dx_{1}\wedge dy_{4}\right)\text{+
}y_{15}\otimes\left(dx_{6}\wedge dy_{7}\right)\end{array}$

(44)
$\tiny\begin{array}[]{ll}\deg=-6&(\alpha+1)\,h_{3}\otimes\left(dx_{9}\right){}^{\wedge
2}\text{+ }\left(\alpha^{2}+\alpha\right)\,x_{2}\otimes\left(dx_{4}\wedge
dx_{13}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,x_{2}\otimes\left(dx_{7}\wedge
dx_{12}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,x_{5}\otimes\left(dx_{4}\wedge
dx_{14}\right)\text{+ }(\alpha+1)\,x_{5}\otimes\left(dx_{9}\wedge
dx_{12}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,x_{8}\otimes\left(dx_{7}\wedge
dx_{14}\right)\\\ &\text{+ }(\alpha+1)\,x_{8}\otimes\left(dx_{9}\wedge
dx_{13}\right)\text{+ }(\alpha+1)\,y_{1}\otimes\left(dx_{7}\wedge
dx_{9}\right)\text{+ }(\alpha+1)\,y_{4}\otimes\left(dx_{6}\wedge
dx_{9}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{4}\otimes\left(dx_{13}\wedge
dy_{2}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{4}\otimes\left(dx_{14}\wedge
dy_{5}\right)\text{+ }(\alpha+1)\,y_{6}\otimes\left(dx_{4}\wedge
dx_{9}\right)\\\ &\text{+ }(\alpha+1)\,y_{7}\otimes\left(dx_{1}\wedge
dx_{9}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{7}\otimes\left(dx_{12}\wedge
dy_{2}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{7}\otimes\left(dx_{14}\wedge
dy_{8}\right)\\\ &\text{+ }(\alpha+1)\,y_{9}\otimes\left(dx_{1}\wedge
dx_{7}\right)\text{+ }(\alpha+1)\,y_{9}\otimes\left(dx_{4}\wedge
dx_{6}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{9}\otimes\left(dx_{12}\wedge
dy_{5}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{9}\otimes\left(dx_{13}\wedge
dy_{8}\right)\text{+ }y_{12}\otimes\left(dx_{7}\wedge dy_{2}\right)\text{+
}y_{12}\otimes\left(dx_{9}\wedge dy_{5}\right)\text{+
}y_{13}\otimes\left(dx_{4}\wedge dy_{2}\right)\\\ &\text{+
}y_{13}\otimes\left(dx_{9}\wedge dy_{8}\right)\text{+
}y_{14}\otimes\left(dx_{4}\wedge dy_{5}\right)\text{+
}y_{14}\otimes\left(dx_{7}\wedge dy_{8}\right)\\\ &\\\
\deg=-6&\left(\alpha^{3}+\alpha^{2}\right)\,x_{1}\otimes\left(dx_{8}\wedge
dx_{10}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,x_{1}\otimes\left(dx_{15}\wedge
dy_{4}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,x_{4}\otimes\left(dx_{2}\wedge
dx_{13}\right)\\\ &\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,x_{4}\otimes\left(dx_{8}\wedge
dx_{11}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,x_{4}\otimes\left(dx_{15}\wedge
dy_{1}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,x_{9}\otimes\left(dx_{3}\wedge
dx_{15}\right)\\\ &\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,x_{9}\otimes\left(dx_{8}\wedge
dx_{13}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{2}\otimes\left(dx_{3}\wedge
dx_{10}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{2}\otimes\left(dx_{13}\wedge
dy_{4}\right)\\\ &\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{3}\otimes\left(dx_{2}\wedge
dx_{10}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{3}\otimes\left(dx_{5}\wedge
dx_{8}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,y_{3}\otimes\left(dx_{15}\wedge
dy_{9}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{5}\otimes\left(dx_{3}\wedge
dx_{8}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{8}\otimes\left(dx_{3}\wedge
dx_{5}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{8}\otimes\left(dx_{10}\wedge
dy_{1}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{8}\otimes\left(d_{11}\wedge
dy_{4}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{8}\otimes\left(dx_{13}\wedge
dy_{9}\right)\text{+ }\alpha\,y_{10}\otimes\left(dx_{2}\wedge
dx_{3}\right)\text{+ }\alpha\,y_{10}\otimes\left(dx_{8}\wedge dy_{1}\right)\\\
&\text{+ }\left(\alpha^{2}+\alpha\right)\,y_{11}\otimes\left(dx_{8}\wedge
dy_{4}\right)\text{+ }\alpha\,y_{13}\otimes\left(dx_{2}\wedge
dy_{4}\right)\text{+ }\alpha\,y_{13}\otimes\left(dx_{8}\wedge
d_{9}\right)\text{+ }y_{15}\otimes\left(dx_{3}\wedge dy_{9}\right)\text{+
}y_{15}\otimes\left(dy_{1}\wedge dy_{4}\right)\\\ &\\\
\deg=-4&(\alpha+1)\,h_{3}\otimes\left(dx_{6}\right){}^{\wedge 2}\text{+
}(\alpha+1)\,h_{4}\otimes\left(dx_{6}\right){}^{\wedge 2}\text{+
}\left(\alpha^{2}+\alpha\right)\,x_{2}\otimes\left(dx_{3}\wedge
dx_{10}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,x_{2}\otimes\left(dx_{13}\wedge
dy_{4}\right)\\\ &\text{+ }(\alpha+1)\,x_{4}\otimes\left(dx_{6}\wedge
dx_{9}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,x_{4}\otimes\left(dx_{13}\wedge
dy_{2}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,x_{4}\otimes\left(dx_{14}\wedge
dy_{5}\right)\\\ &\text{+ }(\alpha+1)\,x_{5}\otimes\left(dx_{6}\wedge
dx_{10}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,x_{5}\otimes\left(dx_{14}\wedge
dy_{4}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,x_{11}\otimes\left(dx_{3}\wedge
dx_{14}\right)\\\ &\text{+ }(\alpha+1)\,x_{11}\otimes\left(dx_{6}\wedge
dx_{13}\right)+(\alpha+1)\,y_{1}\otimes\left(dx_{3}\wedge dx_{6}\right)\text{+
}(\alpha+1)\,y_{3}\otimes\left(dx_{1}\wedge dx_{6}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{3}\otimes\left(dx_{10}\wedge
dy_{2}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{3}\otimes\left(d_{14}\wedge
dy_{11}\right)\text{+ }(\alpha+1)\,y_{6}\otimes\left(dx_{1}\wedge
d_{3}\right)\text{+ }(\alpha+1)\,y_{6}\otimes\left(d_{9}\wedge
dy_{4}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{6}\otimes\left(dx_{10}\wedge
dy_{5}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{6}\otimes\left(dx_{13}\wedge
dy_{11}\right)\text{+ }(\alpha+1)\,y_{9}\otimes\left(dx_{6}\wedge
dy_{4}\right)\text{+ }y_{10}\otimes\left(dx_{3}\wedge dy_{2}\right)\\\
&\text{+ }y_{10}\otimes\left(dx_{6}\wedge dy_{5}\right)\text{+
}y_{13}\otimes\left(dx_{6}\wedge dy_{11}\right)\text{+
}y_{13}\otimes\left(dy_{2}\wedge dy_{4}\right)\text{+
}y_{14}\otimes\left(dx_{3}\wedge dy_{11}\right)\text{+
}y_{14}\otimes\left(dy_{4}\wedge dy_{5}\right)\\\ &\\\
\deg=-4&\left(\alpha^{3}+\alpha^{2}\right)\,h_{4}\otimes\left(dx_{5}\right){}^{\wedge
2}\text{+ }\left(\alpha^{3}+\alpha^{2}\right)\,x_{3}\otimes\left(dx_{2}\wedge
dx_{10}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,x_{3}\otimes\left(dx_{5}\wedge
dx_{8}\right)\\\ &\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,x_{3}\otimes\left(dx_{15}\wedge
dy_{9}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,x_{6}\otimes\left(dx_{5}\wedge
dx_{10}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,x_{6}\otimes\left(dx_{15}\wedge
dy_{7}\right)\\\ &\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,x_{7}\otimes\left(dx_{2}\wedge
dx_{12}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,x_{7}\otimes\left(dx_{5}\wedge
dx_{11}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,x_{7}\otimes\left(dx_{15}\wedge
dy_{6}\right)\\\ &\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,x_{9}\otimes\left(dx_{5}\wedge
dx_{12}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,x_{9}\otimes\left(dx_{15}\wedge
dy_{3}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{2}\otimes\left(dx_{10}\wedge
dy_{3}\right)\\\ &\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,y_{2}\otimes\left(dx_{12}\wedge
dy_{7}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{5}\otimes\left(dx_{8}\wedge
dy_{3}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{5}\otimes\left(dx_{10}\wedge
dy_{6}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{5}\otimes\left(dx_{11}\wedge
dy_{7}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{5}\otimes\left(d(x_{12}\wedge
dy_{9}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{8}\otimes\left(dx_{5}\wedge
dy_{3}\right)\\\ &\text{+ }\alpha\,y_{10}\otimes\left(dx_{2}\wedge
dy_{3}\right)\text{+ }\alpha\,y_{10}\otimes\left(dx_{5}\wedge
dy_{6}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{11}\otimes\left(dx_{5}\wedge
dy_{7}\right)\text{+ }\alpha\,y_{12}\otimes\left(dx_{2}\wedge dy_{7}\right)\\\
&\text{+ }\alpha\,y_{12}\otimes\left(dx_{5}\wedge dy_{9}\right)\text{+
}y_{15}\otimes\left(dy_{3}\wedge dy_{9}\right)\text{+
}y_{15}\otimes\left(dy_{6}\wedge dy_{7}\right)\\\ &\\\
\deg=-2&(\alpha+1)\,h_{4}\otimes\left(dx_{1}\right){}^{\wedge 2}\text{+
}\left(\alpha^{2}+\alpha\right)\,x_{2}\otimes\left(d(x_{10}\wedge
dy_{3}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,x_{2}\otimes\left(dx_{12}\wedge
dy_{7}\right)\text{+ }(\alpha+1)\,x_{3}\otimes\left(dx_{1}\wedge
dx_{6}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,x_{3}\otimes\left(dx_{10}\wedge
dy_{2}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,x_{3}\otimes\left(dx_{14}\wedge
dy_{11}\right)\text{+ }(\alpha+1)\,x_{7}\otimes\left(dx_{1}\wedge
dx_{9}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,x_{7}\otimes\left(dx_{12}\wedge
dy_{2}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,x_{7}\otimes\left(d_{14}\wedge
dy_{8}\right)\text{+ }(\alpha+1)\,x_{8}\otimes\left(dx_{1}\wedge
dx_{10}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,x_{8}\otimes\left(dx_{14}\wedge
dy_{7}\right)\text{+ }(\alpha+1)\,x_{11}\otimes\left(dx_{1}\wedge
dx_{12}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,x_{11}\otimes\left(dx_{14}\wedge
dy_{3}\right)\\\ &\text{+ }(\alpha+1)\,y_{1}\otimes\left(dx_{6}\wedge
dy_{3}\right)\text{+ }(\alpha+1)\,y_{1}\otimes\left(dx_{9}\wedge
dy_{7}\right)\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{1}\otimes\left(dx_{10}\wedge
dy_{8}\right)\\\ &\text{+
}\left(\alpha^{2}+\alpha\right)\,y_{1}\otimes\left(dx_{12}\wedge
dy_{11}\right)\text{+ }(\alpha+1)\,y_{6}\otimes\left(dx_{1}\wedge
dy_{3}\right)\text{+ }(\alpha+1)\,y_{9}\otimes\left(dx_{1}\wedge
dy_{7}\right)\text{+ }y_{10}\otimes\left(dx_{1}\wedge dy_{8}\right)\\\
&\text{+ }y_{10}\otimes\left(dy_{2}\wedge dy_{3}\right)\text{+
}y_{12}\otimes\left(dx_{1}\wedge dy_{11}\right)\text{+
}y_{12}\otimes\left(dy_{2}\wedge dy_{7}\right)\text{+
}y_{14}\otimes\left(dy_{3}\wedge dy_{11}\right)\text{+
}y_{14}\otimes\left(dy_{7}\wedge dy_{8}\right)\\\ &\\\
\deg=0&\alpha\,h_{1}\otimes\left(d_{14}\wedge dy_{14}\right)\text{+
}\alpha^{2}\,h_{1}\otimes\left(dx_{15}\wedge dy_{15}\right)\text{+
}h_{2}\otimes\left(dx_{5}\wedge dy_{5}\right)\text{+
}h_{2}\otimes\left(dx_{8}\wedge dy_{8}\right)\text{+
}h_{2}\otimes\left(dx_{10}\wedge dy_{10}\right)\\\ &\text{+
}h_{2}\otimes\left(dx_{11}\wedge dy_{11}\right)\text{+
}h_{2}\otimes\left(dx_{12}\wedge dy_{12}\right)\text{+
}h_{2}\otimes\left(dx_{13}\wedge dy_{13}\right)\text{+
}\alpha^{2}\,h_{2}\otimes\left(dx_{14}\wedge dy_{14}\right)\text{+
}h_{3}\otimes\left(dx_{1}\wedge dy_{1}\right)\\\ &\text{+
}\alpha\,h_{3}\otimes\left(dx_{5}\wedge dy_{5}\right)\text{+
}h_{3}\otimes\left(dx_{6}\wedge dy_{6}\right)\text{+
}\alpha\,h_{3}\otimes\left(dx_{8}\wedge dy_{8}\right)\text{+
}h_{3}\otimes\left(dx_{9}\wedge dy_{9}\right)\text{+
}\alpha\,h_{3}\otimes\left(dx_{10}\wedge dy_{10}\right)\\\ &\text{+
}\alpha\,h_{3}\otimes\left(dx_{11}\wedge dy_{11}\right)\text{+
}\alpha\,h_{3}\otimes\left(dx_{12}\wedge dy_{12}\right)\text{+
}\left(\alpha^{3}+\alpha^{2}\right)\,h_{3}\otimes\left(dx_{14}\wedge
dy_{14}\right)\text{+
}\left(\alpha^{4}+\alpha^{3}\right)\,h_{3}\otimes\left(dx_{15}\wedge
dy_{15}\right)\\\ &\text{+ }\alpha\,h_{4}\otimes\left(dx_{12}\wedge
dy_{12}\right)\text{+ }\alpha\,h_{4}\otimes\left(dx_{13}\wedge
dy_{13}\right)\text{+ }\alpha\,h_{4}\otimes\left(dx_{14}\wedge
dy_{14}\right)\text{+ }\alpha^{2}\,h_{4}\otimes\left(dx_{15}\wedge
dy_{15}\right)\text{+ }x_{1}\otimes\left(dh_{1}\wedge dx_{1}\right)\\\
&\text{+ }\alpha\,x_{1}\otimes\left(dx_{10}\wedge dy_{8}\right)\text{+
}\alpha\,x_{1}\otimes\left(dx_{12}\wedge dy_{11}\right)\text{+
}\alpha\,x_{1}\otimes\left(dx_{14}\wedge dy_{13}\right)\text{+
}x_{2}\otimes\left(dh_{1}\wedge dx_{2}\right)\text{+
}x_{2}\otimes\left(dx_{5}\wedge dy_{1}\right)\\\ &\text{+
}x_{2}\otimes\left(dx_{8}\wedge dy_{6}\right)\text{+
}x_{2}\otimes\left(dx_{11}\wedge dy_{9}\right)\text{+
}\alpha^{3}\,x_{2}\otimes\left(dx_{15}\wedge dy_{14}\right)\text{+
}\alpha\,x_{3}\otimes\left(dx_{13}\wedge dy_{12}\right)\text{+
}x_{4}\otimes\left(dh_{1}\wedge dx_{4}\right)\\\ &\text{+
}\alpha\,x_{4}\otimes\left(dx_{12}\wedge dy_{10}\right)\text{+
}x_{5}\otimes\left(dx_{10}\wedge dy_{6}\right)\text{+
}x_{5}\otimes\left(dx_{12}\wedge dy_{9}\right)\text{+
}\alpha\,x_{5}\otimes\left(dx_{15}\wedge dy_{13}\right)\text{+
}x_{6}\otimes\left(dh_{1}\wedge dx_{6}\right)\\\ &\text{+
}\alpha\,x_{6}\otimes\left(dx_{10}\wedge dy_{5}\right)\text{+
}\alpha\,x_{6}\otimes\left(dx_{13}\wedge dy_{11}\right)\text{+
}\alpha\,x_{6}\otimes\left(dx_{14}\wedge dy_{12}\right)\text{+
}x_{7}\otimes\left(dh_{1}\wedge dx_{7}\right)\text{+
}\alpha\,x_{7}\otimes\left(dx_{13}\wedge dy_{10}\right)\\\ &\text{+
}x_{8}\otimes\left(dx_{10}\wedge dy_{1}\right)\text{+
}x_{8}\otimes\left(dx_{13}\wedge dy_{9}\right)\text{+
}\alpha\,x_{8}\otimes\left(dx_{15}\wedge dy_{12}\right)\text{+
}\alpha\,x_{9}\otimes\left(dx_{12}\wedge dy_{5}\right)\text{+
}\alpha\,x_{9}\otimes\left(dx_{13}\wedge dy_{8}\right)\\\ &\text{+
}\alpha\,x_{9}\otimes\left(dx_{14}\wedge dy_{10}\right)\text{+
}x_{10}\otimes\left(dh_{1}\wedge dx_{10}\right)\text{+
}x_{10}\otimes\left(dx_{14}\wedge dy_{9}\right)\text{+
}\alpha\,x_{10}\otimes\left(dx_{15}\wedge dy_{11}\right)\text{+
}x_{11}\otimes\left(dh_{1}\wedge dx_{11}\right)\\\ &\text{+
}x_{11}\otimes\left(dx_{12}\wedge dy_{1}\right)\text{+
}x_{11}\otimes\left(dx_{13}\wedge dy_{6}\right)\text{+
}\alpha\,x_{11}\otimes\left(dx_{15}\wedge dy_{10}\right)\text{+
}x_{12}\otimes\left(dx_{14}\wedge dy_{6}\right)\text{+
}\alpha\,x_{12}\otimes\left(dx_{15}\wedge dy_{8}\right)\\\ &\text{+
}x_{13}\otimes\left(dx_{14}\wedge dy_{1}\right)\text{+
}\alpha\,x_{13}\otimes\left(dx_{15}\wedge dy_{5}\right)\text{+
}x_{14}\otimes\left(dh_{1}\wedge dx_{14}\right)\text{+
}y_{1}\otimes\left(dh_{1}\wedge dy_{1}\right)\text{+
}\alpha\,y_{1}\otimes\left(dx_{8}\wedge dy_{10}\right)\\\ &\text{+
}\alpha\,y_{1}\otimes\left(dx_{11}\wedge dy_{12}\right)\text{+
}\alpha\,y_{1}\otimes\left(dx_{13}\wedge dy_{14}\right)\text{+
}y_{2}\otimes\left(dh_{1}\wedge dy_{2}\right)\text{+
}y_{2}\otimes\left(dx_{1}\wedge dy_{5}\right)\text{+
}y_{2}\otimes\left(dx_{6}\wedge dy_{8}\right)\\\ &\text{+
}y_{2}\otimes\left(dx_{9}\wedge dy_{11}\right)\text{+
}\alpha^{3}\,y_{2}\otimes\left(dx_{14}\wedge dy_{15}\right)\text{+
}\alpha\,y_{3}\otimes\left(dx_{12}\wedge dy_{13}\right)\text{+
}y_{4}\otimes\left(dh_{1}\wedge dy_{4}\right)\text{+
}\alpha\,y_{4}\otimes\left(dx_{10}\wedge dy_{12}\right)\\\ &\text{+
}y_{5}\otimes\left(dx_{6}\wedge dy_{10}\right)\text{+
}y_{5}\otimes\left(dx_{9}\wedge dy_{12}\right)\text{+
}\alpha\,y_{5}\otimes\left(dx_{13}\wedge dy_{15}\right)\text{+
}y_{6}\otimes\left(dh_{1}\wedge dy_{6}\right)\text{+
}\alpha\,y_{6}\otimes\left(dx_{5}\wedge dy_{10}\right)\\\ &\text{+
}\alpha\,y_{6}\otimes\left(dx_{11}\wedge dy_{13}\right)\text{+
}\alpha\,y_{6}\otimes\left(dx_{12}\wedge dy_{14}\right)\text{+
}y_{7}\otimes\left(dh_{1}\wedge dy_{7}\right)\text{+
}\alpha\,y_{7}\otimes\left(dx_{10}\wedge dy_{13}\right)\text{+
}y_{8}\otimes\left(dx_{1}\wedge dy_{10}\right)\\\ &\text{+
}y_{8}\otimes\left(dx_{9}\wedge dy_{13}\right)\text{+
}\alpha\,y_{8}\otimes\left(dx_{12}\wedge dy_{15}\right)\text{+
}\alpha\,y_{9}\otimes\left(dx_{5}\wedge dy_{12}\right)\text{+
}\alpha\,y_{9}\otimes\left(dx_{8}\wedge dy_{13}\right)\text{+
}\alpha\,y_{9}\otimes\left(dx_{10}\wedge dy_{14}\right)\\\ &\text{+
}y_{10}\otimes\left(dh_{1}\wedge dy_{10}\right)\text{+
}y_{10}\otimes\left(dx_{9}\wedge dy_{14}\right)\text{+
}\alpha\,y_{10}\otimes\left(dx_{11}\wedge dy_{15}\right)\text{+
}y_{11}\otimes\left(dh_{1}\wedge dy_{11}\right)\text{+
}y_{11}\otimes\left(dx_{1}\wedge dy_{12}\right)\\\ &\text{+
}y_{11}\otimes\left(dx_{6}\wedge dy_{13}\right)\text{+
}\alpha\,y_{11}\otimes\left(dx_{10}\wedge dy_{15}\right)\text{+
}y_{12}\otimes\left(dx_{6}\wedge dy_{14}\right)\text{+
}\alpha\,y_{12}\otimes\left(dx_{8}\wedge dy_{15}\right)\text{+
}y_{13}\otimes\left(dx_{1}\wedge dy_{14}\right)\\\ &\text{+
}\alpha\,y_{13}\otimes\left(dx_{5}\wedge dy_{15}\right)\text{+
}y_{14}\otimes\left(dh_{1}\wedge dy_{14}\right)\end{array}$

###  4.9. ${\mathfrak{gl}}(4)$ and its simple relative

The Lie algebra444Observe that ${\mathfrak{psl}}(4):={\mathfrak{g}}(2)$ in the
basis given in [FH], p. 346. Shen himself observed this, and therefore the
property of the root systems to have all roots of “equal length” (whatever
that might mean if $p=2$) is not an invariant if $p=2$.
${\mathfrak{g}}={\mathfrak{psl}}(4)$ has no Cartan matrix; its relative that
has a Cartan matrix which is $(\ref{gl4cm})$ is ${\mathfrak{gl}}(4)$.

######  Lemma.

In order to compare with Shen’s “variations of ${\mathfrak{g}}(2)$ theme” we
give, in parentheses, together with the generators of ${\mathfrak{sl}}(4)$,
their weights in terms of the root systems of ${\mathfrak{gl}}(4)$ ($\beta$’s)
and ${\mathfrak{g}}(2)$ ($\alpha$’s):

(45) $\begin{pmatrix}{\bar{0}}&1&0\\\ 1&{\bar{0}}&1\\\
0&1&{\bar{0}}\end{pmatrix}\qquad\tiny\begin{array}[]{l}x_{1}=E_{12}\;(\beta_{1}=\alpha_{2}),\;\;x_{2}=E_{23}\;(\beta_{2}=\alpha_{1}),\;\;x_{3}=E_{34}\;(\beta_{3}=2\alpha_{1}+\alpha_{2}),\\\
{}x_{4}:=[x_{1}\text{,$\,$}x_{2}]=E_{13}\;(\beta_{1}+\beta_{2}=\alpha_{1}+\alpha_{2}),\;\;x_{5}:=[x_{2}\text{,$\,$}x_{3}]=E_{24}\;(\beta_{2}+\beta_{3}=3\alpha_{1}+\alpha_{2}),\\\
{}x_{6}:=[x_{3}\text{,$\,$}[x_{1}\text{,$\,$}x_{2}]]=E_{14}\;(\beta_{1}+\beta_{2}+\beta_{3}=3\alpha_{1}+2\alpha_{2})\end{array}$

Then $H^{2}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycles
$(\ref{eqpsl4})$ and $(\ref{eqpsl4bis})$. The parameters $a$, $b$ in
parentheses near the degree correspond to Shen’s parameters of his
$V_{4}{\mathfrak{g}}(2;a,b)$.

(46) $\tiny\begin{array}[]{lll}\deg=-6&-4(\alpha_{1}+\alpha_{2})&y_{1}\otimes
dx_{4}\wedge dx_{6}+y_{4}\otimes dx_{1}\wedge dx_{6}+y_{6}\otimes dx_{1}\wedge
dx_{4}\\\\[8.53581pt] &&\\\ \deg=-6&-4(2\alpha_{1}+\alpha_{2})&y_{3}\otimes
dx_{5}\wedge dx_{6}+y_{5}\otimes dx_{3}\wedge dx_{6}+y_{6}\otimes dx_{3}\wedge
dx_{5}\\\ &&\\\ \deg=-4\;(b)&-2(2\alpha_{1}+\alpha_{2})&h_{2}\otimes
dx_{4}\wedge dx_{5}+y_{2}\otimes dh_{3}\wedge dx_{6}+y_{2}\otimes dx_{1}\wedge
x_{5}+y_{2}\otimes dx_{3}\wedge dx_{4}+y_{4}\otimes dh_{3}\wedge dx_{5}+\\\
&&+y_{5}\otimes dh_{3}\wedge dx_{4}+y_{6}\otimes dh_{3}\wedge
dx_{2}+y_{6}\otimes dx_{4}\wedge dy_{1}+y_{6}\otimes dx_{5}\wedge dy_{3}\\\
&&\\\ \deg=-4&-3(\alpha_{1}+\alpha_{2})&h_{3}\otimes dx_{2}\wedge
dx_{6}+x_{1}\otimes dx_{4}\wedge dx_{6}+x_{3}\otimes dx_{5}\wedge
dx_{6}+y_{2}\otimes dh_{2}\wedge dx_{6}\\\ &&+y_{2}\otimes dh_{3}\wedge
dx_{6}+y_{2}\otimes dx_{1}\wedge dx_{5}+y_{2}\otimes dx_{3}\wedge
dx_{4}+y_{4}\otimes dh_{2}\wedge dx_{5}+\\\ &&y_{4}\otimes dh_{3}\wedge
dx_{5}+y_{4}\otimes dx_{2}\wedge dx_{3}+y_{5}\otimes dh_{2}\wedge
dx_{4}+y_{5}\otimes dh_{3}\wedge dx_{4}\\\ &&+y_{5}\otimes dx_{1}\wedge
dx_{2}+y_{6}\otimes dh_{2}\wedge dx_{2}+y_{6}\otimes dh_{3}\wedge dx_{2}\\\
\end{array}$ (47) $\tiny\begin{array}[]{lll}\deg=-2&0&x_{3}\otimes
dx_{2}\wedge dx_{4}+y_{2}\otimes dx_{4}\wedge dy_{3}+y_{4}\otimes dx_{2}\wedge
dy_{3}\\\ &&\\\ \deg=-2&-4\alpha_{1}&x_{1}\otimes dx_{2}\wedge
dx_{5}+y_{2}\otimes dx_{5}\wedge dy_{1}+y_{5}\otimes dx_{2}\wedge dy_{1}\\\
&&\\\ \deg=-2&-2(\alpha_{1}+\alpha_{2})&h_{3}\otimes dx_{6}\wedge
dy_{2}+x_{2}\otimes dh_{2}\wedge dx_{6}+y_{1}\otimes dh_{2}\wedge
dx_{3}+y_{1}\otimes dx_{5}\wedge dy_{2}+y_{1}\otimes dx_{6}\wedge
dy_{4}\\\\[8.53581pt] &&+y_{3}\otimes dh_{2}\wedge dx_{1}+y_{3}\otimes
dx_{4}\wedge dy_{2}+y_{3}\otimes dx_{6}\wedge dy_{5}+y_{6}\otimes dh_{2}\wedge
dy_{2}\\\ &&\\\ \deg=-2&-2(\alpha_{1}+\alpha_{2})&h_{2}\otimes dx_{1}\wedge
dx_{3}+x_{2}\otimes dh_{3}\wedge dx_{6}+x_{4}\otimes dx_{1}\wedge
dx_{6}+x_{5}\otimes dx_{3}\wedge dx_{6}+y_{1}\otimes dh_{3}\wedge
dx_{3}\\\\[8.53581pt] &&+y_{1}\otimes dx_{5}\wedge dy_{2}+y_{1}\otimes
dx_{6}\wedge dy_{4}+y_{3}\otimes dh_{3}\wedge dx_{1}+y_{3}\otimes dx_{4}\wedge
dy_{2}+y_{3}\otimes dx_{6}\wedge dy_{5}+\\\ &&y_{4}\otimes dx_{3}\wedge
dy_{2}+y_{5}\otimes dx_{1}\wedge dy_{2}+y_{6}\otimes dh_{3}\wedge dy_{2}\\\
&&\\\ \deg=0&2\alpha_{1}&h_{3}\otimes dx_{4}\wedge dy_{5}+x_{3}\otimes
dh_{2}\wedge dx_{1}+x_{3}\otimes dx_{4}\wedge dy_{2}+x_{3}\otimes dx_{6}\wedge
dy_{5}+x_{5}\otimes dh_{2}\wedge dx_{4}\\\\[8.53581pt] &&+y_{1}\otimes
dh_{2}\wedge dy_{3}+y_{1}\otimes dx_{2}\wedge dy_{5}+y_{1}\otimes dx_{4}\wedge
dy_{6}+y_{4}\otimes dh_{2}\wedge dy_{5}\\\ &&\\\
\deg=0&-2\alpha_{1}&h_{3}\otimes dx_{5}\wedge dy_{4}+x_{1}\otimes dh_{2}\wedge
dx_{3}+x_{1}\otimes dx_{5}\wedge dy_{2}+x_{1}\otimes dx_{6}\wedge
dy_{4}+x_{4}\otimes dh_{2}\wedge dx_{5}\\\\[8.53581pt] &&+y_{3}\otimes
dh_{2}\wedge dy_{1}+y_{3}\otimes dx_{2}\wedge dy_{4}+y_{3}\otimes dx_{5}\wedge
dy_{6}+y_{5}\otimes dh_{2}\wedge dy_{4}\\\ &&\\\
\deg=0\;(a)&2\alpha_{1}&h_{2}\otimes dx_{1}\wedge dy_{3}+x_{2}\otimes
dx_{4}\wedge dy_{3}+x_{3}\otimes dh_{2}\wedge dx_{1}+x_{3}\otimes dh_{3}\wedge
dx_{1}+x_{5}\otimes dh_{2}\wedge dx_{4}\\\\[8.53581pt] &&+x_{5}\otimes
dh_{3}\wedge dx_{4}+x_{5}\otimes dx_{1}\wedge dx_{2}+y_{1}\otimes dh_{2}\wedge
dy_{3}+y_{1}\otimes dh_{3}\wedge dy_{3}+y_{1}\otimes dx_{2}\wedge dy_{5}+\\\
&&y_{1}\otimes dx_{4}\wedge dy_{6}+y_{4}\otimes dh_{2}\wedge
dy_{5}+y_{4}\otimes dh_{3}\wedge dy_{5}+y_{4}\otimes dx_{1}\wedge
dy_{6}+y_{6}\otimes dy_{3}\wedge dy_{5}\\\ &&\\\
\deg=0&-2\alpha_{1}&h_{2}\otimes dx_{3}\wedge dy_{1}+x_{1}\otimes dh_{2}\wedge
dx_{3}+x_{1}\otimes dh_{3}\wedge dx_{3}+x_{2}\otimes dx_{5}\wedge
dy_{1}+x_{4}\otimes dh_{2}\wedge dx_{5}\\\\[8.53581pt] &&+x_{4}\otimes
dh_{3}\wedge dx_{5}+x_{4}\otimes dx_{2}\wedge dx_{3}+y_{3}\otimes dh_{2}\wedge
dy_{1}+y_{3}\otimes dh_{3}\wedge dy_{1}+y_{3}\otimes dx_{2}\wedge dy_{4}+\\\
&&y_{3}\otimes dx_{5}\wedge dy_{6}+y_{5}\otimes dh_{2}\wedge
dy_{4}+y_{5}\otimes dh_{3}\wedge dy_{4}+y_{5}\otimes dx_{3}\wedge
dy_{6}+y_{6}\otimes dy_{1}\wedge dy_{4}\end{array}$

##  5\. On integrability

Consider the bracket

(48) $[\cdot,\cdot]_{c}=[\cdot,\cdot]+tc.$

######  Lemma.

For $p=3$, the bracket $(\ref{lindef})$ satisfies the Jacobi identity in the
following cases:

1) For ${\mathfrak{br}}(2)$, ${\mathfrak{brj}}(2,3)$, ${\mathfrak{br}}(2,5)$
and ${\mathfrak{br}}(3)$, for every of the above homogeneous cocycles $c$;

2) For ${\mathfrak{g}}(2,3)$, never;

3) For ${\mathfrak{g}}(1,6)$, only for the cocycle $c$ of degree $-12$.

######  Remark.

Although JI is not satisfied for the bracket (48) for some cocycles $c$, it
does not mean that these cocycles are not integrable: The Jacobi identity
might be satisfied by a bracket of the form
$[\cdot,\cdot]_{c}=[\cdot,\cdot]+tc_{1}+t^{2}c_{2}+\dots$. We have started to
investigate this option elsewhere, together with integrability study of
multiparameter deformations, except for the simplest cases where the global
deform is linear in all parameters simultaneously.

######  Lemma.

For $p=2$, the bracket $(\ref{lindef})$ satisfies the Jacobi identity in the
following cases:

1) For ${\mathfrak{o}}^{(1)}(3)$, ${\mathfrak{o}}^{(1)}(5)$,
${\mathfrak{psl}}(4)$, and ${\mathfrak{psl}}(6)$, for every of the above
homogeneous cocycles $c$.

2) For ${\mathfrak{wk}}(3;\alpha)$ and ${\mathfrak{wk}}(4;\alpha)$, and also
for ${\mathfrak{bgl}}(3;\alpha)$ and ${\mathfrak{bgl}}(4;\alpha)$, for every
of the above homogeneous cocycles $c$, except the ones of degree $0$.

5) For ${\mathfrak{o}}{\mathfrak{o}}_{I\Pi}^{(1)}(1|2)$, for every of the
above homogeneous cocycles $c$.

######  Lemma.

For $p=2$, the Lie algebra ${\mathfrak{o}}^{(1)}_{I}(3)$ admits the global
deformation given by

$[\cdot,\cdot]_{\alpha,\beta}=[\cdot,\cdot]+\alpha c_{1}+\beta c_{2}.$

Denote the deform by ${\mathfrak{o}}^{(1)}_{I}(3,\alpha,\beta)$. This Lie
algebra is simple if and only if $\alpha\beta\not=1$.

If $\alpha\beta=1$, then ${\mathfrak{o}}^{(1)}_{I}(3,\alpha,\beta)$ has an
ideal $I=\mathop{\mathrm{{Span}}}\nolimits\\{h,x+\alpha y\\}$.

For $\alpha\beta\not=1$, the Lie algebra
${\mathfrak{o}}^{(1)}_{I}(3,\alpha,\beta)$ is simple with two outer
derivations given by

$\alpha h\otimes dh+\alpha x\otimes dx+x\otimes dy,\quad\beta h\otimes
dh+\beta x\otimes dx+y\otimes dx.$

Proof of the statements of this Lemma are straightforward.

Reducing the operator $\mathop{\mathrm{ad}}\nolimits_{h}$ to the Jordan normal
form, we immediately see (in the eigenbasis) that the deform depends,
actually, on one parameter, not two; and all of them are non-isomorphic.

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