# Derivations and central extensions of simple modular Lie algebras and
superalgebras

Sofiane Bouarroudj1, Pavel Grozman2, Alexei Lebedev2, Dimitry Leites3 1New
York University Abu Dhabi, Division of Science and Mathematics, P.O. Box
129188, United Arab Emirates<EMAIL_ADDRESS>
2Equa Simulation AB, Stockholm, Sweden<EMAIL_ADDRESS><EMAIL_ADDRESS>
3Department of Mathematics, Stockholm University, Roslagsv. 101, Kräftriket
hus 6, SE-106 91 Stockholm, Sweden<EMAIL_ADDRESS>
###### Abstract.

Over algebraically closed fields of positive characteristic, we list the outer
derivations and non-trivial central extensions of simple finite dimensional
Lie algebras and Lie superalgebras of the rank sufficiently small to allow
computer-aided study using Mathematica-based code SuperLie. We consider
algebras possessing a grading by integers such that the dimensions of the
superspaces whose degrees are equal in absolute value coincide (symmetric
algebras), and also non-trivial deforms of these algebras. When the answer is
clear for the infinite series, it is given for any rank.

We also consider the non-symmetric case of periplectic Lie superalgebras and
the Lie algebras obtained from them by desuperization.

###### Key words and phrases:

modular Lie superalgebra

###### 1991 Mathematics Subject Classification:

17B50, 70F25

S.B. was partly supported by the grant AD 065 NYUAD

##  1\. Introduction

Hereafter, ${\mathbb{K}}$ is an algebraically closed field of characteristic
$p>0$ and ${\mathfrak{g}}$ is a finite dimensional Lie (super)algebra.

###  1.1. Goal: classify simple modular Lie algebras

This major problem was attacked from the two directions.

####  1.1.1. All algebras, but $p>5$.

In late 1960s, Kostrikin and Shafarevich formulated a conjecture describing a
method producing all simple finite dimensional restricted modular Lie algebras
over algebraically closed fields of characteristic $p>7$, and a generalization
embracing non-restricted algebras. The statement of the conjecture turned out
to hold for $p=7$ as well. The Kostrikin-Shafarevich procedure for obtaining
all simple finite dimensional Lie algebras consists of the following steps:

(1) 1) considering the two types of simple complex Lie algebras: a) those of
the form ${\mathfrak{g}}(A)$ for a Cartan matrix $A$, b) infinite dimensional
vectorial Lie algebras with polynomial coefficients; 2) selecting certain
bases allowing integer structure constants (Chevalley bases for the algebras
of the form ${\mathfrak{g}}(A)$ and divided powers for vectorial Lie algebras)
thus getting ${\mathbb{Z}}$-forms of these complex Lie algebras; 3) tensoring
the ${\mathbb{Z}}$-forms obtained at step 2) by ${\mathbb{K}}$ over
${\mathbb{Z}}$; 4) selecting a simple (and finite dimensional in the vectorial
Lie algebra case) subquotient, called a simple “relative” in what follows; 5)
deforming the results obtained at step 4); 6) classification of isomorphisms
between Lie algebras obtained at earlier steps.

At first, addressing this classification problem (goal), Shafarevich, together
with his student, Kostrikin, only considered restricted 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 indication to look at the groups (geometry) rather
than at Lie algebras if $p>0$. Since only restricted Lie algebras correspond
to groups we interpret Deligne’s advice as a certain natural indication to
restrict the classification problem of simple Lie (super)algebras to
restricted ones at the first step.

####  1.1.2. All algebras, $p=5$.

For $p=5$, Melikyan found a series of algebras not produced by the method
suggested by Kostrikin and Shafarevich. More examples not fitting the KSh-
method were known for $p=3$ and 2.

Summing up 30 years of work of several teams of researchers, Block and 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 $p>3$; for reviews, see [S, BGP].

####  1.1.3. Arbitrary $p$, but algebras only with indecomposable Cartan
matrix, or simple subquotient thereof.

For a particular case of finite dimensional Lie algebras ${\mathfrak{g}}(A)$
with indecomposable Cartan matrix $A$ for any $p>0$, Weisfeiler and Kac [WK]
offered a classification.

Although the idea of the proof in [WK] is OK, the paper has several gaps and
vague notions: the Brown algebra ${\mathfrak{br}}(3)$, see [Br3] was missed;
the notion of the Lie algebra with Cartan matrix nicely defined in [K] was not
properly developed at the time [WK] was written; the quotients
${\mathfrak{g}}(A)^{(i)}/{\mathfrak{c}}$ of the first or second (for $i=1$ or
2) derived modulo center ${\mathfrak{c}}$ have no Cartan matrix but were often
referred as having one. All the needed preliminary notions are carefully
defined in [BGL1] where the result of [WK] with Skryabin’s correction (he
observed that ${\mathfrak{br}}(3)$ possesses a Cartan matrix, actually, two
inequivalent ones) is double-checked.

####  1.1.4. A conjectural method for producing all simple Lie algebras for
$p\geq 3$.

For its exposition, see [Ltow].

The main procedure for constructing simple Lie (super)algebras in the
conjecture [Ltow], that claims to embrace $p=2$ as well, is the _generalized
Cartan prolongation_.

The main characters in this procedure are simple Lie (super)algebras, their
central extensions and algebras of derivations, and modules over such
algebras.

The process is inductive: Given a simple Lie (super)algebra, we describe its
“relatives”, i.e., central extensions and algebra of derivations; they might
be building bricks in the construction of a new simple algebra. In various
problems these relatives of simple Lie (super)algebra are no less important
than simple algebras.

####  1.1.5. Simple algebras and their relatives with important properties.
Outer derivations and central extensions.

For the Lie (super)algebras with an (even) supersymmetric invariant bilinear
form $b$, the problems of description of these two objects (outer derivations
and non-trivial central extensions) are almost equivalent. Indeed, the outer
derivations are described by cocycles representing the classes of
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ whereas the non-trivial central
extensions are defined by cocycles representing the classes of
$H^{2}({\mathfrak{g}})$, i.e., with trivial coefficients. In presence of such
a form $b$, we have ${\mathfrak{g}}\simeq{\mathfrak{g}}^{*}$, and hence
$H^{1}({\mathfrak{g}};{\mathfrak{g}})\simeq
H^{1}({\mathfrak{g}};{\mathfrak{g}}^{*})$.

There are maps $i:H^{1}({\mathfrak{g}};{\mathfrak{g}}^{*})\longrightarrow
H^{2}({\mathfrak{g}})$ and $j:H^{2}({\mathfrak{g}})\longrightarrow
H^{1}({\mathfrak{g}};{\mathfrak{g}}^{*})$ given, for any cocycle $f$
representing a class $[f]\in H^{1}({\mathfrak{g}};{\mathfrak{g}}^{*})$, any
cocycle $\omega$ representing a class $[\omega]\in H^{2}({\mathfrak{g}})$, and
$X,Y\in{\mathfrak{g}}$, by the formulas

(2) $\begin{array}[]{l}i(f)(X,Y)=f(X)(Y),\\\
j(\omega)(X)(Y)=\omega(X,Y).\end{array}$

The relation between these cohomology spaces is given by the following exact
sequence

(3) $0\longrightarrow H^{2}({\mathfrak{g}})\stackrel{{\scriptstyle
j}}{{\longrightarrow}}H^{1}({\mathfrak{g}};{\mathfrak{g}}^{*})\stackrel{{\scriptstyle
v}}{{\longrightarrow}}(S^{2}({\mathfrak{g}}))^{\mathfrak{g}}\stackrel{{\scriptstyle
K}}{{\longrightarrow}}H^{3}({\mathfrak{g}})\longrightarrow
H^{2}({\mathfrak{g}};{\mathfrak{g}}^{*})\longrightarrow
H^{1}({\mathfrak{g}};S^{2}({\mathfrak{g}})),$

where $(S^{2}({\mathfrak{g}}))^{\mathfrak{g}}$ denotes the space of symmetric
bilinear ${\mathfrak{g}}$-invariant forms on ${\mathfrak{g}}$, and the maps
$v$ and $K$ are defined as follows:

$\begin{array}[]{ll}v(d)(X,Y)=d(X)(Y)+(-1)^{p(X)p(X)}d(Y)(X)&\text{for any
$X,Y\in{\mathfrak{g}}$ and $d\in
Z^{1}({\mathfrak{g}};{\mathfrak{g}}^{*})$},\\\
K(b)(X,Y,Z)=b([X,Y],Z)&\text{for any $X,Y,Z\in{\mathfrak{g}}$ and
$b\in(S^{2}({\mathfrak{g}}))^{\mathfrak{g}}$}.\end{array}$

The beginning of the exact sequence (3) was discovered by Koszul, who
introduced the map $K$, the exact sequence was further extended to the right
on different occasions, see [NW, DzZ]. The fact that the map $j$ is inclusion
is proved in [Dz5]; the exactness of the sequence (3) is proved in [NW],
Proposition 7.2. For the case of _even_ invariant symmetric bilinear forms
$(S^{2}({\mathfrak{g}}))^{\mathfrak{g}}$, the arguments of [NW] are literally
applicable to Lie superalgebras.

“What is the relation between $H^{2}({\mathfrak{g}})$ and
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ if ${\mathfrak{g}}$ possesses an _odd_
symmetric ${\mathfrak{g}}$-invariant bilinear form?” is an open question.
Anyway, it is clear that if $(S^{2}({\mathfrak{g}}))^{\mathfrak{g}}=0$, there
is an isomorphism $H^{2}({\mathfrak{g}})\simeq
H^{1}({\mathfrak{g}};{\mathfrak{g}}^{*})$; this is wonderful, except that we
are interested in $H^{1}({\mathfrak{g}};{\mathfrak{g}})$, instead. So we have
to compute this space, anyway. Or, the other way round, information about
outer derivations does not help to say what are the central extensions, and we
have to compute $H^{2}({\mathfrak{g}})$, anyway. There is, nevertheless,
something we can say a priori (and double-check the results).

####  1.1.6. What can be said a priori.

Let ${\mathfrak{g}}={\mathfrak{g}}(A)$ be a Lie (super)algebra with a
degenerate Cartan matrix. Then ${\mathfrak{g}}$ has a central element $C$ and
a grading element $D$, both lying in the maximal torus ${\mathfrak{h}}$ and
$D\not\in{\mathfrak{g}}^{\prime}$. Assume, for simplicity, that
$\mathop{\mathrm{rk}}\nolimits(A)={\text{size}}(A)-1$, so
${\mathfrak{g}}={\mathfrak{g}}^{\prime}\mathop{{\subset}\hbox
to0.0pt{\hss\raise 0.5pt\hbox{\small+}\hskip 0.5pt}}{\mathbb{K}}\cdot D$, a
semidirect sum with ${\mathfrak{g}}^{\prime}$ an ideal.

######  Statement.

The operator $M$ on ${\mathfrak{g}}$ corresponding to the cochain $C\otimes
d(D)$, i.e., such that

(4) $MD=C;\quad M{\mathfrak{g}}^{\prime}=0,$

is an outer derivation of ${\mathfrak{g}}$.

###### Proof.

To show that it is a derivation, we need to show that

$M[x,y]=[Mx,y]+[x,My]\ \text{for any $x,y\in{\mathfrak{g}}$}.$

But the left-hand side is 0 thanks to (4), and the right-hand side is 0 due to
the fact that $\mathop{\text{\rm Im}}\nolimits M\subset{\mathfrak{c}}$, where
${\mathfrak{c}}$ is the center of ${\mathfrak{g}}$.

This derivation is an outer one, i.e., there is no $x\in{\mathfrak{g}}$ such
that $M=\mathop{\mathrm{ad}}\nolimits_{x}$. Indeed, otherwise $[x,D]=C$, but
$D\in{\mathfrak{h}}$, so $[D,{\mathfrak{h}}]=0$;
$[D,{\mathfrak{g}}_{\pm}]\subset{\mathfrak{g}}_{\pm}$. Thus,
$C\not\in[D,{\mathfrak{g}}]$ since $C\in{\mathfrak{h}}$. ∎

If ${\text{size}}(A)-\mathop{\mathrm{rk}}\nolimits(A)=k$, then there are $k$
linearly independent central elements $C_{1},\dots,C_{k}$, and $k$ grading
elements $D_{1},\dots,D_{k}$. Similarly to the above, there are $k^{2}$
linearly independent outer derivations of ${\mathfrak{g}}$, i.e., operators
$M_{ij}$ corresponding to the cochains $C_{i}\otimes d(D_{j})$.

####  1.1.7. When $p=2$ problems and obstacles pile up. Hidden
supersymmetries.

It was clear since mid-1970s that to classify simple Lie algebras in
characteristic 2, we have to incorporate Lie superalgebras: forgetting
superstructure (“desuperizing”) makes any Lie superalgebra into a Lie algebra;
this process often preserves simplicity.

The other way round, “hidden supersymmetries” of Lie algebras, i.e.,
possibility to obtain Lie superalgebra with the same “family name” as a given
Lie algebra, e.g., ${\mathfrak{gl}}$ or ${\mathfrak{e}}$, for ANY (both simple
and “relative” thereof) Lie algebra.

Further cogitations on the case of $p=2$ and elucidations of the basics made
it clear (to A. Lebedev, this was a Chapter in his Ph.D. thesis, [LeD]) that
the existence of a 2-structure on a simple Lie algebra leads to a new type of
simple Lie superalgebras, the type never noticed before: _queerifications of
restricted simple Lie algebras_.

Since the lack of outer derivations is a sufficient condition for the the
algebra to be restricted, it was natural to describe these derivations for the
cases not considered in [FG], and — since this description is of independent
interest — for Lie superalgebras. This is what we do in what follows for all
“symmetric” Lie (super)algebras of small rank, and in several general cases
(with high probability for all cases in these dimensions, not just for the
“known” ones, thanks to the results by Eick [Ei]).

For simple ${\mathbb{Z}}$-graded vectorial Lie algebras, non-trivial central
extensions are described mainly by A. Dzhumadildaev, see [Dz4, DzU].

For “lopsided” Lie algebras not considered by Dzhumadildaev, and “lopsided”
Lie superalgebras, (we have in mind only simple ones and “relatives” thereof),
to compute the outer derivatives and non-trivial central extensions is an open
problem.

###  1.2. Super goal: classify simple modular Lie superalgebras

####  1.2.1. Lie superalgebras with indecomposable Cartan matrices, their
simple subquotients.

These are classified in [BGL1]. If $p=2$, there is one more type of symmetric
Lie algebras, the orthogonal Lie algebras. Among the derived of the orthogonal
Lie algebras, there is also an important version of “lopsided” Lie algebras,
an analog of the periplectic Lie superalgebras. One more type of the symmetric
Lie superalegbras is produced by restricted simple Lie algebras of the form
${\mathfrak{g}}(A)$ or their subquotients.

The deforms of the symmetric Lie (super)algebras are studied in [BGL2],
results are partial.

All other types of simple Lie superalgebras are not classified yet. There are
only available conjectural methods for producing all of them, see [Ltow].

####  1.2.2. Simple vectorial Lie superalgebras.

For $p>5$, one can, conjecturally, use same Kostrikin-Shafarevich procedure
(1) with the following simple Lie superalgebras over ${\mathbb{C}}$ for the
input: of the form ${\mathfrak{g}}(A)$ (classified for various types of growth
in [Ser, HS], see also [CCLL]); simple vectorial with polynomial coefficients
(classified in [LSh]), together with the special periplectic
${\mathfrak{spe}}(n)$ and “queer” Lie superalgebras ${\mathfrak{psq}}(n)$
(first discovered in [Dj]), and pass to simple subquotients of all the above.

For $p=5$, the same conjecture is, perhaps, true; minus Melikyan algebras and
plus examples of the form ${\mathfrak{g}}(A)$ and related subquotients, see
[BGL1, BGLL].

For $p=3$, there appear new examples: For algebras, see [GL4]; for
superalgebras, see [BL, BGL3].

For $p=2$, there appear still new examples: For algebras, see [GL4]; for
superalgebras, see [LeP, BGLLSq].

##  2\. Main results

In this section we collected the description of outer derivations.

One of the main types or our examples are Lie (super)algebras with Cartan
matrix, or their “relatives”. Of course, the weights of cocycles do not depend
on a realization (the choice of Cartan matrix) but the form of cocycles does,
so for each Lie (super)algebra, we give the result for one (“simplest” in some
sense) incarnation if there are several. We indicate the Cartan matrix to
which our cocycles correspond. Thanks to symmetry, it suffices to give only
cocycles of non-positive degree. We give only the “non-super” version of the
Cartan matrix; for Lie superalgebras with the “same” Cartan matrix (but with 0
instead of ${\bar{0}}$ on the diagonal) the form of the cocycle is virtually
the same, so we do not list these super cases separately, bar several
exceptional cases where the answer is distinct (for reasons unknown).[ DL: ]

The answer for the simple subquotient ${\mathfrak{g}}^{(1)}(A)/{\mathfrak{c}}$
of the Lie (super)algebra of the form ${\mathfrak{g}}(A)$ is given with
respect to the same basis $x$ for positive root vectors and $y$ for negative
root vectors as for ${\mathfrak{g}}(A)$ itself. The elements $h$ belong to the
maximal torus: $h_{i}:=[x_{i},y_{i}]$; for the singular Cartan matrices, we
denote the elements $d_{j}$, see [BGL1], also by letters $h$ in order not to
confuse them with the differential $d$. The following cocycles are labeled by
their degree defined by setting $\deg x_{i}=-\deg y_{i}=1$ for every Chevalley
generator (do not confuse generators with the other elements of the Chevalley
basis); the superscript numerates cocycles of the same degree, if any such
occur.

Let
${\mathfrak{out}}{\mathfrak{g}}:={\mathfrak{aut}}({\mathfrak{g}})/{\mathfrak{g}}$
be the Lie (super)algebra of outer derivations of ${\mathfrak{g}}$.

We consider algebras as rank grows, but if the general answer for a series
considered is clear, we give the answer at the first instance.

###  2.1. Rank=1

####  2.1.1. $p=2$.

######  Lemma.

For ${\mathfrak{o}}^{(1)}_{\Pi}(2n+1)$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by $2n$ cocycles of weight
$-2n$, $-2n+2$, …, $2n-2$, $2n$ and
${\mathfrak{aut}}{\mathfrak{g}}={\mathfrak{o}}_{\Pi}(2n+1)$.

######  Lemma.

For ${\mathfrak{hei}}(2)$, we have
${\mathfrak{out}}{\mathfrak{g}}\simeq{\mathfrak{gl}}(2)$; and the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycles:

$\begin{array}[]{ll}c_{-2}=&y_{1}\otimes dx_{1}\\\ c_{0}^{1}=&h_{1}\otimes
dh_{1}+x_{1}\otimes dx_{1}\\\ c_{0}^{2}=&x_{1}\otimes dx_{1}+y_{1}\otimes
dy_{1}\end{array}$

######  Remark.

The result is correct, though not symmetric. Let us do it by hand. The basis
of the space of 1-cochains of degree 0 is $\\{h_{1}\otimes dh_{1},\
x_{1}\otimes dx_{1},\ y_{1}\otimes dy_{1}\\}$. If we compute the differential
of a linear combination of these cocycles with coefficients $i_{s}$ for
$s=1,2,3$, we get

(5) $(i_{1}+i_{2}+i_{3})h_{1}\otimes dx_{1}\wedge dy_{1}\ .$

For the linear combination to be a cocycle, the expression (5) should vanish,
i.e., $i_{1}+i_{2}+i_{3}=0$. Besides, there are now coboundaries of degree 0
because, $d(h_{1})=0$. Now, _SuperLie_ is taking the basis
$i_{1}=i_{2}=1,i_{3}=0$ and $i_{1}=0,i_{2}=i_{3}=1$. To have a symmetric
answer, we’d take the basis with $i_{1}=i_{3}=1,i_{2}=0$ and
$i_{1}=i_{2}=1,i_{3}=0$; the code _SuperLie_ has different esthetic criteria.
We encounter the same type of answer for $p=3$.

######  Lemma.

For ${\mathfrak{hei}}(0|2)$, we have
${\mathfrak{out}}{\mathfrak{g}}\simeq{\mathfrak{o}}_{\Pi}(2)\oplus{\mathbb{K}}z$;
the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
cocycles for $p=2$, $3$, $5$, …, and $0$:

$\begin{array}[]{l}c_{0}^{1}=h_{1}\otimes dh_{1}-x_{1}\otimes dx_{1}\\\
c_{0}^{2}=-x_{1}\otimes dx_{1}+y_{1}\otimes dy_{1}\end{array}$

######  Lemma.

For ${\mathfrak{o}}{\mathfrak{o}}_{I\Pi}(1|2)$, we have
$H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

######  Remark.

One might expect the same result as for ${\mathfrak{o}}^{(1)}_{\Pi}(3)$. Lemma
2.1.1d. Lemma is, however, correct: let us do it by hand again. There is one
coboundary: $x_{1}\otimes dx_{1}+y_{1}\otimes dy_{1}$. The space of cochains
is 5-dimensional, that of cocycles is equal to 2. Now for
${\mathfrak{o}}^{(1)}_{\Pi}(3)$, the situation is different (first, the
dimension is smaller), the space of cochains is 3-dimensional, and that of
cocycles is equal to 1. The coboundary, which is also $x_{1}\otimes
dx_{1}+y_{1}\otimes dy_{1}$, reduces to 0. We encounter the same type of
answer for $p=3$.

####  2.1.2. $p=3$.

######  Lemma.

For ${\mathfrak{hei}}(2)$, we have
${\mathfrak{out}}{\mathfrak{g}}\simeq{\mathfrak{gl}}(2)$; the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycles

$\begin{array}[]{ll}c_{-2}=&y_{1}\otimes dx_{1}\\\ c_{0}^{1}=&h_{1}\otimes
dh_{1}+\ x_{1}\otimes dx_{1}\\\ c_{0}^{2}=&-x_{1}\otimes dx_{1}+y_{1}\otimes
dy_{1}\end{array}$

######  Lemma.

For ${\mathfrak{osp}}(1|2)$, the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$
is spanned by the following cocycles (here $x_{2}=[x_{1},x_{1}]$ and the
$y_{2}$ is similarly defined):

$\begin{array}[]{ll}c_{-3}=&2\ y_{1}\otimes dx_{2}+y_{2}\otimes
dx_{1}\end{array}$

######  Lemma.

For ${\mathfrak{sl}}(2)$, we have $H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

####  2.1.3. $p=5$.

######  Lemma.

For ${\mathfrak{hei}}(2)$, we have
${\mathfrak{out}}{\mathfrak{g}}\simeq{\mathfrak{gl}}(2)$; the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycles:

$\begin{array}[]{ll}c_{-2}=&y_{1}\otimes dx_{1}\\\ c_{0}^{1}=&h_{1}\otimes
dh_{1}-x_{1}\otimes dx_{1}\\\ c_{0}^{2}=&-x_{1}\otimes dx_{1}+y_{1}\otimes
dy_{1}\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{sl}}(2)$ and ${\mathfrak{osp}}(1|2)$, we have
$H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

###  2.2. Rank=2

####  2.2.1. $p=2$.

######  Lemma.

For ${\mathfrak{o}}{\mathfrak{o}}_{I\Pi}^{(1)}(3|2)$ with Cartan matrix

(6) $\begin{pmatrix}0&1\\\ 1&1\end{pmatrix}\quad\text{and the basis
}\;\begin{array}[]{l}x_{1},\ x_{2},\\\ x_{3}=x_{2}^{2},x_{4}=[x_{1},x_{2}],\\\
x_{5}=[x_{2},x_{2}^{2}],\end{array}$

the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
cocycles:

$\begin{array}[]{ll}c_{-4}=&y_{2}\otimes dx_{5}+y_{4}\otimes
dx_{4}+y_{5}\otimes dx_{2}\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{sl}}(3)$, ${\mathfrak{sl}}(1|2)$,
${\mathfrak{o}}{\mathfrak{o}}_{I\Pi}^{(1)}(1|2)$,
${\mathfrak{o}}{\mathfrak{o}}_{I\Pi}^{(1)}(1|4)$, we have
$H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

####  2.2.2. $p=3$.

######  Lemma.

a) For ${\mathfrak{g}}={\mathfrak{gl}}(3)$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycle:

$c_{0}=2\ h_{1}\otimes dh_{3}+h_{2}\otimes dh_{3}$

Above $h_{3}$ denotes the grading operator in order not to confuse it with the
differential $d$.

b) For ${\mathfrak{g}}={\mathfrak{psl}}(3)$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycles:

${\footnotesize\begin{array}[]{ll}c_{-3}=&y_{1}\otimes dx_{3}+x_{3}\otimes
dx_{1}+y_{2}\otimes dx_{3}+x_{3}\otimes dx_{2}\\\\[8.53581pt] c_{0}^{1}=&2\
x_{2}\otimes dx_{1}+y_{1}\otimes dy_{2}\\\ c_{0}^{2}=&x_{1}\otimes dx_{1}+2\
x_{2}\otimes dx_{2}+2\ y_{1}\otimes dy_{1}+y_{2}\otimes dy_{2}\\\
c_{0}^{3}=&2\ x_{1}\otimes dx_{2}+y_{2}\otimes dy_{1}\end{array}}$

######  Remark.

This remark is for those who can not believe the above answer in heading a)
since “${\mathfrak{gl}}$ can not have outer derivations by definition”. In a
way, the expression in quotation marks is not true if $p\neq 0$. Indeed, the
Cartan matrix of ${\mathfrak{gl}}(3)$ for $p=3$ is not invertible, so there is
a grading operator $d$, see [BGL1]. For such a $d$ one can take, i.e.,
$h_{3}=2E_{1,1}+E_{2,2}+E_{3,3}$, which, of course, belongs to the algebra.

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{brj}}(2;3)$ with Cartan matrix

(7) $\begin{pmatrix}0&-1\\\ -2&1\end{pmatrix}\quad\text{and the basis
}{\footnotesize\;\begin{array}[]{l}x_{1},\;\;x_{2},\\\ x_{3}=[x_{1},x_{2}],\
x_{4}=[x_{2},x_{2}],\\\ x_{5}=[x_{2},[x_{1},x_{2}]],\\\
x_{6}=[[x_{1},x_{2}],[x_{2},x_{2}]],\\\
x_{7}=[[x_{2},x_{2}],[x_{2},[x_{1},x_{2}]]],\\\
x_{8}=[[x_{2},[x_{1},x_{2}]],[x_{2},[x_{1},x_{2}]]],\end{array}}$

the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
cocycles:

$c_{-3}=x_{1}\otimes dx_{6}+x_{3}\otimes dx_{7}+2\ y_{2}\otimes
dx_{4}+y_{4}\otimes dx_{2}+2\ y_{6}\otimes dy_{1}+y_{7}\otimes dy_{3}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{br}}(3)$, ${\mathfrak{sl}}(1|2)$,
${\mathfrak{o}}(5)$, ${\mathfrak{osp}}(3|2)$, ${\mathfrak{osp}}(1|4)$, we have
$H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

####  2.2.3. $p=5$.

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{brj}}(2;5)$, ${\mathfrak{sl}}(3)$,
${\mathfrak{o}}(5)$, ${\mathfrak{sl}}(1|2)$, ${\mathfrak{osp}}(1|2)$, and
${\mathfrak{osp}}(1|4)$, we have $H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

###  2.3. Rank=3

####  2.3.1. $p=2$.

######  Lemma.

a) For ${\mathfrak{g}}={\mathfrak{gl}}(4)$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycle
(here $h_{4}$ is the outer derivation; same situation as for
${\mathfrak{gl}}(3)$ for $p=3$):

$c_{0}=h_{1}\otimes dh_{4}+h_{3}\otimes dh_{4}$

b) For ${\mathfrak{g}}={\mathfrak{psl}}(4)$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycles
and the corresponding subalgebra of ${\mathfrak{aut}}{\mathfrak{g}}$ is
indicated on the right if we can identify it:

${\footnotesize\begin{array}[]{ll|l}c_{-4}=&y_{2}\otimes dx_{6}+y_{4}\otimes
dx_{5}+y_{5}\otimes dx_{4}+y_{6}\otimes dx_{2}&{\bf
F}({\mathfrak{h}}(0|4))\\\\[8.53581pt] c_{-2}=&x_{2}\otimes
dx_{6}+y_{1}\otimes dx_{3}+y_{3}\otimes dx_{1}+y_{6}\otimes
dy_{2}\\\\[8.53581pt] c_{0}^{1}=&x_{3}\otimes dx_{1}+x_{5}\otimes
dx_{4}+y_{1}\otimes dy_{3}+y_{4}\otimes dy_{5}\\\ c_{0}^{2}=&x_{1}\otimes
dx_{3}+x_{4}\otimes dx_{5}+y_{3}\otimes dy_{1}+y_{5}\otimes dy_{4}\\\
c_{0}^{3}=&x_{2}\otimes dx_{2}+x_{3}\otimes dx_{3}+x_{4}\otimes
dx_{4}+y_{2}\otimes dy_{2}+&\\\ &y_{3}\otimes dy_{3}+y_{4}\otimes
dy_{4}&{\mathfrak{pgl}}(4)\simeq{\bf
F}({\mathfrak{h}}^{\prime}(0|4)\mathop{{\subset}\hbox to0.0pt{\hss\raise
0.5pt\hbox{\small+}\hskip 0.5pt}}{\mathbb{K}}E)\end{array}}$

The non-trivial central extensions of ${\mathfrak{psl}}(4)$: the right-most
column contains the description of the result when we can identify the
algebra; the independent cocycles are as follows:

(8) $\begin{array}[]{ll|l}c_{-4}=&dx_{2}\wedge dx_{6}+dx_{4}\wedge dx_{5}&{\bf
F}({\mathfrak{po}}^{\prime}(0|4))\\\ c_{-2}=&dx_{1}\wedge dx_{3}+dx_{6}\wedge
dy_{2}\\\ c_{0}^{1}=&dx_{1}\wedge dy_{3}+dx_{4}\wedge dy_{5}\\\
c_{0}^{2}=&dx_{3}\wedge dy_{1}+dx_{5}\wedge dy_{4}\\\ c_{0}^{3}=&dx_{2}\wedge
dy_{2}+dx_{3}\wedge dy_{3}+dx_{4}\wedge dy_{4}&{\mathfrak{sl}}(4)\end{array}$

######  Lemma.

a) For ${\mathfrak{g}}={\mathfrak{wk}}(3,\alpha)$ and
${\mathfrak{g}}={\mathfrak{bgl}}(3,\alpha)$ with Cartan matrix

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

the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
cocycle:

$c_{0}=h_{1}\otimes dh_{4}+\alpha\ h_{3}\otimes dh_{4}$

b) For ${\mathfrak{g}}={\mathfrak{wk}}^{(1)}(3,a)/{\mathfrak{c}}$ and
${\mathfrak{g}}={\mathfrak{bgl}}^{(1)}(3,a)/{\mathfrak{c}}$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycle:

$c_{0}=x_{1}\otimes dx_{1}+x_{3}\otimes dx_{3}+x_{4}\otimes
dx_{4}+x_{5}\otimes dx_{5}+y_{1}\otimes dy_{1}+y_{3}\otimes
dy_{3}+y_{4}\otimes dy_{4}+y_{5}\otimes dy_{5}$

####  2.3.2. $p=3$.

######  Lemma.

a) For ${\mathfrak{g}}={\mathfrak{gl}}(2|2)$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
cocycle(here $h_{4}$ is the outer derivation; same situation as for
${\mathfrak{gl}}(3)$ for $p=3$):

$c_{0}=2\ h_{1}\otimes dh_{4}+h_{2}\otimes dh_{4}+h_{3}\otimes dh_{4}$

b) For ${\mathfrak{g}}={\mathfrak{psl}}(2|2)$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycle:

$c_{0}=2x_{3}dx_{3}+2x_{4}dx_{4}+2x_{5}dx_{5}+y_{3}dy_{3}+y_{4}dy_{4}+y_{5}dy_{5},\
{\mathfrak{aut}}{\mathfrak{g}}={\mathfrak{pgl}}(2|2).$

######  Lemma.

a) For ${\mathfrak{g}}={\mathfrak{g}}(2,3)$ with Cartan matrix

(10) $\begin{pmatrix}2&-1&-1\\\ -1&2&-1\\\ -1&-1&0\end{pmatrix}\text{ and
basis }{\footnotesize\begin{array}[]{l}x_{1},\;\;x_{2},\;\;x_{3},\\\
x_{4}=[x_{1},x_{2}],\;\;x_{5}=[x_{1},x_{3}],\;\;x_{6}=[x_{2},x_{3}],\\\
x_{7}=[x_{3},[x_{1},x_{2}]],\\\
x_{8}=[[x_{1},x_{2}],[x_{1},x_{3}]],\;\;x_{9}=[[x_{1},x_{2}],[x_{2},x_{3}]],\\\
x_{10}=[[x_{1},x_{2}],[x_{3},[x_{1},x_{2}]]],\\\
x_{11}=[[x_{3},[x_{1},x_{2}]],[x_{3},[x_{1},x_{2}]]]\end{array}}$

the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
cocycle:

$c_{0}=2\ h_{1}\otimes dh_{4}+h_{2}\otimes dh_{4}$

b) For
${\mathfrak{g}}={\mathfrak{bj}}:={\mathfrak{g}}(2,3)^{(1)}/{\mathfrak{c}}$,
the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
cocycles:

$\begin{array}[]{ll}c_{-3}=&2\ x_{3}\otimes dx_{8}+x_{6}\otimes
dx_{10}+y_{1}\otimes dx_{4}+y_{4}\otimes dx_{1}+2\ y_{8}\otimes
dy_{3}+y_{10}\otimes dy_{6}\\\ c_{-3}=&x_{3}\otimes dx_{9}+x_{5}\otimes
dx_{10}+2\ y_{2}\otimes dx_{4}+2\ y_{4}\otimes dx_{2}+y_{9}\otimes
dy_{3}+y_{10}\otimes dy_{5}\\\\[8.53581pt] c_{0}^{1}=&x_{2}\otimes dx_{1}+2\
x_{6}\otimes dx_{5}+2\ x_{9}\otimes dx_{8}+2\ y_{1}\otimes dy_{2}+y_{5}\otimes
dy_{6}+y_{8}\otimes dy_{9}\\\ c_{0}^{2}=&x_{1}\otimes dx_{1}+2\ x_{2}\otimes
dx_{2}+2\ x_{5}\otimes dx_{5}+x_{6}\otimes dx_{6}+2\ x_{8}\otimes
dx_{8}+x_{9}\otimes dx_{9}\\\ &+2\ y_{1}\otimes dy_{1}+y_{2}\otimes
dy_{2}+y_{5}\otimes dy_{5}+2\ y_{6}\otimes dy_{6}+y_{8}\otimes dy_{8}+2\
y_{9}\otimes dy_{9}\\\ c_{0}^{3}=&x_{1}\otimes dx_{2}+2\ x_{5}\otimes
dx_{6}+2\ x_{8}\otimes dx_{9}+2\ y_{2}\otimes dy_{1}+y_{6}\otimes
dy_{5}+y_{9}\otimes dy_{8}\end{array}$

######  Lemma.

For
${\mathfrak{g}}={\mathfrak{g}}(1,6),{\mathfrak{osp}}(1|6),{\mathfrak{osp}}(2|4),{\mathfrak{osp}}(3|4),{\mathfrak{osp}}(5|2),{\mathfrak{sl}}(1|3),{\mathfrak{osp}}(4|2)$,
we have $H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

####  2.3.3. $p=5$.

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{ag}}(2)$, ${\mathfrak{osp}}(1|6)$,
${\mathfrak{osp}}(2|4)$, ${\mathfrak{osp}}(3|4)$, ${\mathfrak{osp}}(5|2)$,
${\mathfrak{sl}}(1|3)$, ${\mathfrak{osp}}(4|2)$, we have
$H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

######  Lemma.

a) For ${\mathfrak{g}}={\mathfrak{gl}}(2|2)$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycle
(see comment on $h_{4}$ above):

$c_{0}=4\ h_{1}\otimes dh_{4}+3\ h_{2}\otimes dh_{4}+h_{3}\otimes dh_{4}$

b) For ${\mathfrak{g}}={\mathfrak{psl}}(2|2)$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycle:

$c_{0}=4x_{3}d_{3}+4x_{4}dx_{4}+4x_{5}dx_{5}+y_{3}dy_{3}+y_{4}dy_{4}+y_{5}dy_{5},\
{\mathfrak{aut}}{\mathfrak{g}}={\mathfrak{pgl}}(2|2).$

###  2.4. Rank=4

####  2.4.1. $p=2$.

######  Lemma.

(b) For
${\mathfrak{g}}={\mathfrak{o}}{\mathfrak{c}}(1;8)\mathop{{\subset}\hbox
to0.0pt{\hss\raise 0.5pt\hbox{\small+}\hskip 0.5pt}}{\mathbb{K}}I_{0}$,
${\mathfrak{o}}{\mathfrak{o}}{\mathfrak{c}}(1;4|4)\oplus{\mathbb{K}}I_{0}$ and
${\mathfrak{o}}{\mathfrak{o}}{\mathfrak{c}}(1;6|2)/{\mathfrak{c}}$ with Cartan
matrix

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

the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
cocycles:

$\begin{array}[]{l}c_{0}^{1}=h_{1}\otimes dh_{5}+h_{2}\otimes dh_{5}\\\
c_{0}^{2}=h_{1}\otimes dh_{6}+h_{2}\otimes dh_{6}\\\ c_{0}^{3}=h_{1}\otimes
dh_{5}+h_{3}\otimes dh_{5}\\\ c_{0}^{4}=h_{1}\otimes dh_{6}+h_{3}\otimes
dh_{6}\end{array}$

(b) For ${\mathfrak{g}}={\mathfrak{o}}{\mathfrak{c}}(1;8)/{\mathfrak{c}}$,
${\mathfrak{o}}{\mathfrak{o}}{\mathfrak{c}}(1;4|4)/{\mathfrak{c}}$ and
${\mathfrak{o}}{\mathfrak{o}}{\mathfrak{c}}(1;6|2)/{\mathfrak{c}}$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycles:

$\begin{array}[]{ll}c_{-6}^{1}=&y_{1}\otimes dx_{12}+y_{5}\otimes
dx_{11}+y_{8}\otimes dx_{9}+y_{9}\otimes dx_{8}+y_{11}\otimes
dx_{5}+y_{12}\otimes dx_{1}\\\ c_{-6}^{2}=&y_{2}\otimes dx_{12}+y_{6}\otimes
dx_{11}+y_{8}\otimes dx_{10}+y_{10}\otimes dx_{8}+y_{11}\otimes
dx_{6}+y_{12}\otimes dx_{2}\\\ c_{-6}^{3}=&y_{3}\otimes dx_{12}+y_{7}\otimes
dx_{11}+y_{9}\otimes dx_{10}+y_{10}\otimes dx_{9}+y_{11}\otimes
dx_{7}+y_{12}\otimes dx_{3}\\\\[8.53581pt] c_{-4}^{1}=&x_{1}\otimes
dx_{12}+y_{4}\otimes dx_{10}+y_{6}\otimes dx_{7}+y_{7}\otimes
dx_{6}+y_{10}\otimes dx_{4}+y_{12}\otimes dy_{1}\\\ c_{-4}^{2}=&x_{2}\otimes
dx_{12}+y_{4}\otimes dx_{9}+y_{5}\otimes dx_{7}+y_{7}\otimes
dx_{5}+y_{9}\otimes dx_{4}+y_{12}\otimes dy_{2}\\\ c_{-4}^{3}=&x_{3}\otimes
dx_{12}+y_{4}\otimes dx_{8}+y_{5}\otimes dx_{6}+y_{6}\otimes
dx_{5}+y_{8}\otimes dx_{4}+y_{12}\otimes dy_{3}\\\\[8.53581pt]
c_{-2}^{1}=&x_{4}\otimes dx_{10}+x_{5}\otimes dx_{11}+y_{2}\otimes
dx_{3}+y_{3}\otimes dx_{2}+y_{10}\otimes dy_{4}+y_{11}\otimes dy_{5}\\\
c_{-2}^{2}=&x_{4}\otimes dx_{9}+x_{6}\otimes dx_{11}+y_{1}\otimes
dx_{3}+y_{3}\otimes dx_{1}+y_{9}\otimes dy_{4}+y_{11}\otimes dy_{6}\\\
c_{-2}^{3}=&x_{4}\otimes dx_{8}+x_{7}\otimes dx_{11}+y_{1}\otimes
dx_{2}+y_{2}\otimes dx_{1}+y_{8}\otimes dy_{4}+y_{11}\otimes
dy_{7}\\\\[8.53581pt] c_{0}^{1}=&x_{3}\otimes dx_{2}+x_{7}\otimes
dx_{6}+x_{9}\otimes dx_{8}+y_{2}\otimes dy_{3}+y_{6}\otimes
dy_{7}+y_{8}\otimes dy_{9}\\\ c_{0}^{2}=&x_{3}\otimes dx_{1}+x_{7}\otimes
dx_{5}+x_{10}\otimes dx_{8}+y_{1}\otimes dy_{3}+y_{5}\otimes
dy_{7}+y_{8}\otimes dy_{10}\\\ c_{0}^{3}=&x_{2}\otimes dx_{3}+x_{6}\otimes
dx_{7}+x_{8}\otimes dx_{9}+y_{3}\otimes dy_{2}+y_{7}\otimes
dy_{6}+y_{9}\otimes\text{dy}_{8}\\\ c_{0}^{4}=&x_{2}\otimes
dx_{1}+x_{6}\otimes dx_{5}+x_{10}\otimes dx_{9}+y_{1}\otimes
dy_{2}+y_{5}\otimes dy_{6}+y_{9}\otimes dy_{10}\\\ c_{0}^{5}=&x_{1}\otimes
dx_{3}+x_{5}\otimes dx_{7}+x_{8}\otimes dx_{10}+y_{3}\otimes
dy_{1}+y_{7}\otimes dy_{5}+y_{10}\otimes dy_{8}\\\ c_{0}^{6}=&x_{1}\otimes
dx_{2}+x_{5}\otimes dx_{6}+x_{9}\otimes dx_{10}+y_{2}\otimes
dy_{1}+y_{6}\otimes dy_{5}+y_{10}\otimes dy_{9}\\\ c_{0}^{7}=&x_{1}\otimes
dx_{1}+x_{2}\otimes dx_{2}+x_{5}\otimes dx_{5}+x_{6}\otimes
dx_{6}+x_{9}\otimes dx_{9}+x_{10}\otimes dx_{10}+\\\ &y_{1}\otimes
dy_{1}+y_{2}\otimes dy_{2}+y_{5}\otimes dy_{5}+y_{6}\otimes
dy_{6}+y_{9}\otimes dy_{9}+y_{10}\otimes dy_{10}\\\ c_{0}^{8}=&x_{1}\otimes
dx_{1}+x_{3}\otimes dx_{3}+x_{5}\otimes dx_{5}+x_{7}\otimes
dx_{7}+x_{8}\otimes dx_{8}+x_{10}\otimes dx_{10}+\\\ &y_{1}\otimes
dy_{1}+y_{3}\otimes dy_{3}+y_{5}\otimes dy_{5}+y_{7}\otimes
dy_{7}+y_{8}\otimes dy_{8}+y_{10}\otimes dy_{10}\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{o}}{\mathfrak{o}}_{I\Pi}^{(1)}(3|6)$ with
Cartan matrix

(12) $\begin{pmatrix}{\bar{0}}&1&0&0\\\ 1&{\bar{0}}&1&0\\\ 0&1&0&1\\\
0&0&1&{\bar{1}}\end{pmatrix}\text{ and basis
}{\footnotesize\begin{array}[]{l}x_{1},\ x_{2},\ x_{3},\ x_{4},\\\
x_{5}=[x_{1},x_{2}],\ x_{6}=[x_{2},x_{3}],\ x_{7}=[x_{3},x_{4}],\\\
x_{8}=[x_{3},[x_{1},x_{2}]],\ x_{9}=[x_{4},[x_{2},x_{3}]],\
x_{10}=[x_{4},[x_{3},x_{4}]],\\\ x_{11}=[x_{4},[x_{4},[x_{2},x_{3}]]],\
x_{12}=[[x_{1},x_{2}],[x_{3},x_{4}]],\\\ x_{13}=[x_{3},x_{4}]{}^{2},\
x_{14}=[[x_{1},x_{2}],[x_{4},[x_{3},x_{4}]]],\\\
x_{15}=[[x_{3},x_{4}],[x_{4},[x_{2},x_{3}]]],\\\
x_{16}=[[x_{3},[x_{1},x_{2}]],[x_{4},[x_{3},x_{4}]]],\\\
x_{17}=[x_{4},[x_{2},x_{3}]]{}^{2},\
x_{18}=[[x_{4},[x_{2},x_{3}]],[[x_{1},x_{2}],[x_{3},x_{4}]]],\\\
x_{19}=[[x_{1},x_{2}],[x_{3},x_{4}]]{}^{2}\end{array}}$

the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
cocycles:

$\begin{array}[]{ll}c_{-2}=&x_{3}\otimes dx_{10}+x_{6}\otimes
dx_{11}+x_{8}\otimes dx_{14}+y_{4}\otimes dx_{4}+y_{10}\otimes
dy_{3}+y_{11}\otimes dy_{6}+y_{14}\otimes dy_{8}\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{o}}{\mathfrak{o}}_{I\Pi}^{(1)}(5|4)$ with
Cartan matrix

(13) $\begin{pmatrix}{\bar{0}}&1&0&0\\\ 1&0&1&0\\\ 0&1&{\bar{0}}&1\\\
0&0&1&{\bar{1}}\end{pmatrix}\text{ and basis
}{\footnotesize\begin{array}[]{l}x_{1},\ x_{2},\ x_{3},\ x_{4},\\\
x_{5}=[x_{1},x_{2}],\ x_{6}=[x_{2},x_{3}],\ x_{7}=[x_{3},x_{4}],\\\
x_{8}=[x_{3},[x_{1},x_{2}]],\ x_{9}=[x_{4},[x_{2},x_{3}]],\
x_{10}=[x_{4},[x_{3},x_{4}]],\\\ x_{11}=[x_{4},[x_{4},[x_{2},x_{3}]]],x\
_{12}=[[x_{1},x_{2}],[x_{3},x_{4}]],\\\
x_{13}=[[x_{1},x_{2}],[x_{4},[x_{3},x_{4}]]],\
x_{14}=[[x_{3},x_{4}],[x_{4},[x_{2},x_{3}]]],\\\
x_{15}=[[x_{3},[x_{1},x_{2}]],[x_{4},[x_{3},x_{4}]]],\\\
x_{16}=[x_{4},[x_{2},x_{3}]]{}^{2},\
x_{17}=[[x_{4},[x_{2},x_{3}]],[[x_{1},x_{2}],[x_{3},x_{4}]]],\\\
x_{18}=[[x_{1},x_{2}],[x_{3},x_{4}]]{}^{2}\end{array}}$

the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
cocycles:

$\begin{array}[]{ll}c_{-4}=&x_{2}\otimes dx_{14}+x_{5}\otimes
dx_{15}+y_{3}\otimes dx_{10}+y_{7}\otimes dx_{7}+y_{10}\otimes
dx_{3}+y_{14}\otimes dy_{2}+y_{15}\otimes dy_{5}\\\ c_{-2}=&x_{3}\otimes
dx_{10}+x_{6}\otimes dx_{11}+x_{8}\otimes dx_{13}+y_{4}\otimes
dx_{4}+y_{10}\otimes dy_{3}+y_{11}\otimes dy_{6}+y_{13}\otimes
dy_{8}\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{o}}{\mathfrak{o}}_{I\Pi}^{(1)}(7|2)$, with
Cartan matrix

(14) $\begin{pmatrix}0&1&0&0\\\ 1&{\bar{0}}&1&0\\\ 0&1&{\bar{0}}&1\\\
0&0&1&1\end{pmatrix}\text{ and basis }{\footnotesize\begin{array}[]{l}x_{1},\
x_{2},\ x_{3},\ x_{4},\\\ x_{5}=[x_{1},x_{2}],\ x_{6}=[x_{2},x_{3}],\
x_{7}=[x_{3},x_{4}],\\\ x_{8}=[x_{3},[x_{1},x_{2}]],\
x_{9}=[x_{4},[x_{2},x_{3}]],x_{10}=[x_{4},[x_{3},x_{4}]],\\\
x_{11}=[x_{4},[x_{4},[x_{2},x_{3}]]],\
x_{12}=[[x_{1},x_{2}],[x_{3},x_{4}]],\\\
x_{13}=[[x_{1},x_{2}],[x_{4},[x_{3},x_{4}]]],\
x_{14}=[[x_{3},x_{4}],[x_{4},[x_{2},x_{3}]]],\\\
x_{15}=[[x_{3},[x_{1},x_{2}]],[x_{4},[x_{3},x_{4}]]],\\\
x_{16}=[[x_{4},[x_{2},x_{3}]],[[x_{1},x_{2}],[x_{3},x_{4}]]],\\\
x_{17}=[[x_{1},x_{2}],[x_{3},x_{4}]]{}^{2}\end{array}}$

The space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
cocycles:

$\begin{array}[]{ll}c_{-6}=&x_{1}\otimes dx_{16}+y_{2}\otimes
dx_{14}+y_{6}\otimes dx_{11}+y_{9}\otimes dx_{9}+y_{11}\otimes
dx_{6}+y_{14}\otimes dx_{2}+y_{16}\otimes dy_{1}\\\ c_{-4}=&x_{2}\otimes
dx_{14}+x_{5}\otimes dx_{15}+y_{3}\otimes dx_{10}+y_{7}\otimes
dx_{7}+y_{10}\otimes dx_{3}+y_{14}\otimes dy_{2}+y_{15}\otimes dy_{5}\\\
c_{-2}=&x_{3}\otimes dx_{10}+x_{6}\otimes dx_{11}+x_{8}\otimes
dx_{13}+y_{4}\otimes dx_{4}+y_{10}\otimes dy_{3}+y_{11}\otimes
dy_{6}+y_{13}\otimes dy_{8}\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{o}}{\mathfrak{o}}^{(1)}_{I\Pi}(1|8)$,
${\mathfrak{sl}}(1|4)$, ${\mathfrak{sl}}(3|2)$, ${\mathfrak{sl}}(5)$,
${\mathfrak{wk}}(4,a)$, we have $H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

####  2.4.2. $p=3$.

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{g}}(4,3)$, ${\mathfrak{g}}(3,6)$, we have
$H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

######  Lemma.

a) For ${\mathfrak{g}}={\mathfrak{g}}(3,3)$, with the following Cartan matrix
and basis

(15) $\begin{pmatrix}2&-1&0&0\\\ -1&2&-1&0\\\ 0&-2&2&-1\\\
0&0&1&0\end{pmatrix}\ {\footnotesize\begin{array}[]{l}x_{1},\ x_{2},\ x_{3},\
x_{4},\\\ x_{5}=[x_{1},x_{2}],\ x_{6}=[x_{2},x_{3}],\ x_{7}=[x_{3},x_{4}],\\\
x_{8}=[x_{3},[x_{1},x_{2}]],x_{9}=[x_{3},[x_{2},x_{3}]],x_{10}=[x_{4},[x_{2},x_{3}]],\\\
x_{11}=[x_{3},[x_{3},[x_{1},x_{2}]]],x_{12}=[[x_{1},x_{2}],[x_{3},x_{4}]],x_{13}=[[x_{2},x_{3}],[x_{3},x_{4}]],\\\
x_{14}=[[x_{2},x_{3}],[x_{3},[x_{1},x_{2}]]],x_{15}=[[x_{3},x_{4}],[x_{3},[x_{1},x_{2}]]],\\\
x_{16}=[[x_{3},[x_{1},x_{2}]],[x_{4},[x_{2},x_{3}]]],\\\
x_{17}=[[x_{4},[x_{2},x_{3}]],[x_{3},[x_{3},[x_{1},x_{2}]]]]\end{array}}$

the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
cocycle:

$\begin{array}[]{l}c_{0}=2\ h_{1}\otimes dh_{5}+h_{2}\otimes
dh_{5}+h_{4}\otimes dh_{5}\end{array}$

b) For ${\mathfrak{g}}={\mathfrak{g}}(3,3)^{(1)}/{\mathfrak{c}}$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycles:

$\footnotesize\begin{array}[]{ll}c_{-8}=&y_{4}\otimes dx_{17}+y_{7}\otimes
dx_{16}+2\ y_{10}\otimes dx_{15}+y_{12}\otimes dx_{13}+y_{13}\otimes
dx_{12}+2\ y_{15}\otimes dx_{10}+y_{16}\otimes dx_{7}+y_{17}\otimes dx_{4}\\\
c_{0}=&2\ x_{4}\otimes dx_{4}+2\ x_{7}\otimes dx_{7}+2\ x_{10}\otimes
dx_{10}+2\ x_{12}\otimes dx_{12}+2\ x_{13}\otimes dx_{13}+2\ x_{15}\otimes
dx_{15}+\\\ &2\ x_{16}\otimes dx_{16}+2\ x_{17}\otimes dx_{17}+y_{4}\otimes
dy_{4}+y_{7}\otimes dy_{7}+y_{10}\otimes dy_{10}+y_{12}\otimes
dy_{12}+y_{13}\otimes dy_{13}+\\\ &y_{15}\otimes dy_{15}+y_{16}\otimes
dy_{16}+y_{17}\otimes dy_{17}\end{array}$

###  2.5. Rank=5

####  2.5.1. $p=2$.

######  Lemma.

(a) For
${\mathfrak{g}}={\mathfrak{o}}{\mathfrak{c}}(1;10)\mathop{{\subset}\hbox
to0.0pt{\hss\raise 0.5pt\hbox{\small+}\hskip 0.5pt}}{\mathbb{K}}I_{0}$,
${\mathfrak{o}}{\mathfrak{o}}{\mathfrak{c}}(1;8|2)\mathop{{\subset}\hbox
to0.0pt{\hss\raise 0.5pt\hbox{\small+}\hskip 0.5pt}}{\mathbb{K}}I_{0}$, and
${\mathfrak{o}}{\mathfrak{o}}{\mathfrak{c}}(1;6|4)\mathop{{\subset}\hbox
to0.0pt{\hss\raise 0.5pt\hbox{\small+}\hskip 0.5pt}}{\mathbb{K}}I_{0}$, as
well as ${\mathfrak{g}}=\mathfrak{pec}(1;5)\mathop{{\subset}\hbox
to0.0pt{\hss\raise 0.5pt\hbox{\small+}\hskip 0.5pt}}{\mathbb{K}}I_{0}$, with
the following Cartan matrix and basis

(16) $\begin{pmatrix}{\bar{0}}&0&1&0&0\\\ 0&{\bar{0}}&1&0&0\\\
1&1&{\bar{0}}&1&0\\\ 0&0&1&{\bar{0}}&1\\\ 0&0&0&1&{\bar{0}}\end{pmatrix}\
{\footnotesize\begin{array}[]{l}x_{1},x_{2},x_{3},x_{4},x_{5},\\\
x_{6}=[x_{1},x_{3}],x_{7}=[x_{2},x_{3}],x_{8}=[x_{3},x_{4}],x_{9}=[x_{4},x_{5}],\\\
x_{10}=[x_{2},[x_{1},x_{3}]],x_{11}=[x_{4},[x_{1},x_{3}]],x_{12}=[x_{4},[x_{2},x_{3}]],x_{13}=[x_{5},[x_{3},x_{4}]],\\\
x_{14}=[x_{4},[x_{2},[x_{1},x_{3}]]],x_{15}=[[x_{1},x_{3}],[x_{4},x_{5}]],x_{16}=[[x_{2},x_{3}],[x_{4},x_{5}]],\\\
x_{17}=[[x_{3},x_{4}],[x_{2},[x_{1},x_{3}]]],x_{18}=[[x_{4},x_{5}],[x_{2},[x_{1},x_{3}]]],\\\
x_{19}=[[x_{2},[x_{1},x_{3}]],[x_{5},[x_{3},x_{4}]]],\\\
x_{20}=[[x_{5},[x_{3},x_{4}]],[x_{4},[x_{2},[x_{1},x_{3}]]]]\end{array}}$

the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
cocycle:

$\begin{array}[]{ll}c_{0}=h_{1}\otimes dh_{6}+h_{2}\otimes
dh_{6}\par\end{array}$

(b) For ${\mathfrak{g}}={\mathfrak{o}}(10)/{\mathfrak{c}}$,
${\mathfrak{o}}{\mathfrak{o}}(8|2)/{\mathfrak{c}}$, and
${\mathfrak{o}}{\mathfrak{o}}(6|4)/{\mathfrak{c}}$, as well as
${\mathfrak{g}}=\mathfrak{pe}(5)/{\mathfrak{c}}$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycles:

$\begin{array}[]{ll}c_{-8}=&y_{5}\otimes dx_{20}+y_{9}\otimes
dx_{19}+y_{13}\otimes dx_{18}+y_{15}\otimes dx_{16}+y_{16}\otimes
dx_{15}+y_{18}\otimes dx_{13}+y_{19}\otimes dx_{9}+y_{20}\otimes
dx_{5}\\\\[8.53581pt] c_{-6}=&x_{5}\otimes dx_{20}+y_{4}\otimes
dx_{17}+y_{8}\otimes dx_{14}+y_{11}\otimes dx_{12}+y_{12}\otimes
dx_{11}+y_{14}\otimes dx_{8}+y_{17}\otimes dx_{4}+y_{20}\otimes
dy_{5}\\\\[8.53581pt] c_{-4}=&x_{4}\otimes dx_{17}+x_{9}\otimes
dx_{19}+y_{3}\otimes dx_{10}+y_{6}\otimes dx_{7}+y_{7}\otimes
dx_{6}+y_{10}\otimes dx_{3}+y_{17}\otimes dy_{4}+y_{19}\otimes
dy_{9}\\\\[8.53581pt] c_{-2}=&x_{3}\otimes dx_{10}+x_{8}\otimes
dx_{14}+x_{13}\otimes dx_{18}+y_{1}\otimes dx_{2}+y_{2}\otimes
dx_{1}+y_{10}\otimes dy_{3}+y_{14}\otimes dy_{8}+y_{18}\otimes
dy_{13}\\\\[8.53581pt] c_{0}^{1}=&x_{2}\otimes dx_{1}+x_{7}\otimes
dx_{6}+x_{12}\otimes dx_{11}+x_{16}\otimes dx_{15}+y_{1}\otimes
dy_{2}+y_{6}\otimes dy_{7}+y_{11}\otimes dy_{12}+y_{15}\otimes dy_{16}\\\
c_{0}^{2}=&x_{1}\otimes dx_{2}+x_{6}\otimes dx_{7}+x_{11}\otimes
dx_{12}+x_{15}\otimes dx_{16}+y_{2}\otimes dy_{1}+y_{7}\otimes
dy_{6}+y_{12}\otimes dy_{11}+y_{16}\otimes dy_{15}\\\ c_{0}^{3}=&x_{2}\otimes
dx_{2}+x_{4}\otimes dx_{4}+x_{7}\otimes dx_{7}+x_{8}\otimes
dx_{8}+x_{9}\otimes dx_{9}+x_{10}\otimes dx_{10}+x_{11}\otimes
dx_{11}+x_{13}\otimes dx_{13}+\\\ &x_{15}\otimes dx_{15}+x_{20}\otimes
dx_{20}+y_{2}\otimes dy_{2}+y_{4}\otimes dy_{4}+y_{7}\otimes
dy_{7}+y_{8}\otimes dy_{8}+y_{9}\otimes dy_{9}+y_{10}\otimes dy_{10}+\\\
&y_{11}\otimes dy_{11}+y_{13}\otimes dy_{13}+y_{15}\otimes
dy_{15}+y_{20}\otimes dy_{20}\end{array}$

######  Lemma.

(a) For ${\mathfrak{g}}={\mathfrak{gl}}(6)$, as well as
${\mathfrak{g}}={\mathfrak{gl}}(5|1)$, ${\mathfrak{gl}}(4|2)$,
${\mathfrak{gl}}(3|3)$, the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is
spanned by the following cocycle (here $h_{6}$ is the outer derivation; same
situation as for ${\mathfrak{gl}}(3)$ for $p=3$):

$c_{0}=h_{1}\otimes dh_{6}+h_{3}\otimes dh_{6}+h_{5}\otimes dh_{6}$

(b) For ${\mathfrak{g}}={\mathfrak{psl}}(6)$, as well as
${\mathfrak{psl}}(5|1)$, ${\mathfrak{psl}}(4|2)$, ${\mathfrak{psl}}(3|3)$, the
space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
cocycle turning ${\mathfrak{psl}}$ into ${\mathfrak{pgl}}$:

$\begin{array}[]{ll}c_{0}=&x_{3}\otimes dx_{3}+x_{4}\otimes
dx_{4}+x_{7}\otimes dx_{7}+x_{9}\otimes dx_{9}+x_{10}\otimes
dx_{10}+y_{3}\otimes dy_{3}+\\\ &y_{4}\otimes dy_{4}+y_{7}\otimes
dy_{7}+y_{9}\otimes dy_{9}+y_{10}\otimes dy_{10}\end{array}$

######  Lemma.

(a) For ${\mathfrak{g}}={\mathfrak{o}}{\mathfrak{o}}^{(1)}_{I\Pi}(9|2)$,
${\mathfrak{o}}{\mathfrak{o}}^{(1)}_{I\Pi}(7|4)$,
${\mathfrak{o}}{\mathfrak{o}}^{(1)}_{I\Pi}(5|6)$, and
${\mathfrak{o}}{\mathfrak{o}}^{(1)}_{I\Pi}(3|8)$ with the following Cartan
matrix and basis

(17) $\begin{pmatrix}{\bar{0}}&1&0&0&0\\\ 1&0&1&0&0\\\ 0&1&{\bar{0}}&1&0\\\
0&0&1&{\bar{0}}&1\\\
0&0&0&1&{\bar{1}}\end{pmatrix}\quad{\tiny\begin{array}[]{l}x_{1},\ x_{2},\
x_{3},\ x_{4},\ x_{5},\\\ x_{6}=[x_{1},x_{2}],\ x_{7}=[x_{2},x_{3}],\
x_{8}=[x_{3},x_{4}],\ x_{9}=[x_{4},x_{5}],\\\ x_{10}=[x_{3},[x_{1},x_{2}]],\
x_{11}=[x_{4},[x_{2},x_{3}]],\ x_{12}=[x_{5},[x_{3},x_{4}]],\\\ \hskip
14.22636ptx_{13}=[x_{5},[x_{4},x_{5}]],\\\
x_{14}=[x_{5},[x_{5},[x_{3},x_{4}]]],\ x_{15}=[[x_{1},x_{2}],[x_{3},x_{4}]],\
x_{16}=[[x_{2},x_{3}],[x_{4},x_{5}]],\\\
x_{17}=[[x_{2},x_{3}],[x_{5},[x_{4},x_{5}]]],\
x_{18}=[[x_{4},x_{5}],[x_{3},[x_{1},x_{2}]]],\\\ \hskip
14.22636ptx_{19}=[[x_{4},x_{5}],[x_{5},[x_{3},x_{4}]]],\\\
x_{20}=[[x_{3},[x_{1},x_{2}]],[x_{5},[x_{4},x_{5}]]],x_{21}=[[x_{4},[x_{2},x_{3}]],[x_{5},[x_{4},x_{5}]]],\\\
x_{22}=[[x_{5},[x_{3},x_{4}]],[[x_{2},x_{3}],[x_{4},x_{5}]]],\\\ \hskip
14.22636ptx_{23}=[[x_{5},[x_{4},x_{5}]],[[x_{1},x_{2}],[x_{3},x_{4}]]],\\\
x_{24}=[[x_{5},[x_{5},[x_{3},x_{4}]]],[[x_{1},x_{2}],[x_{3},x_{4}]]],\
x_{25}=[[x_{2},x_{3}],[x_{4},x_{5}]]{}^{2},\\\
x_{26}=[[[x_{2},x_{3}],[x_{4},x_{5}]],[[x_{4},x_{5}],[x_{3},[x_{1},x_{2}]]]],\\\
x_{27}=[[x_{4},x_{5}],[x_{3},[x_{1},x_{2}]]]{}^{2}\end{array}}$

the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
cocycles:

${\tiny\begin{array}[]{ll}c_{-8}=&x_{1}\otimes dx_{25}+y_{2}\otimes
dx_{22}+y_{7}\otimes dx_{21}+y_{11}\otimes dx_{17}+y_{16}\otimes
dx_{16}+y_{17}\otimes dx_{11}\\\ &+y_{21}\otimes dx_{7}+y_{22}\otimes
dx_{2}+y_{25}\otimes dy_{1}\\\ c_{-6}=&x_{2}\otimes dx_{22}+x_{6}\otimes
dx_{24}+y_{3}\otimes dx_{19}+y_{8}\otimes dx_{14}+y_{12}\otimes
dx_{12}+y_{14}\otimes dx_{8}\\\ &+y_{19}\otimes dx_{3}+y_{22}\otimes
dy_{2}+y_{24}\otimes dy_{6}\\\ c_{-4}=&x_{3}\otimes dx_{19}+x_{7}\otimes
dx_{21}+x_{10}\otimes dx_{23}+y_{4}\otimes dx_{13}+y_{9}\otimes
dx_{9}+y_{13}\otimes dx_{4}\\\ &+y_{19}\otimes dy_{3}+y_{21}\otimes
dy_{7}+y_{23}\otimes dy_{10}\\\ c_{-2}=&x_{4}\otimes dx_{13}+x_{8}\otimes
dx_{14}+x_{11}\otimes dx_{17}+x_{15}\otimes dx_{20}+y_{5}\otimes
dx_{5}+y_{13}\otimes dy_{4}\\\ &+y_{14}\otimes dy_{8}+y_{17}\otimes
dy_{11}+y_{20}\otimes dy_{15}\end{array}}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{o}}{\mathfrak{o}}^{(1)}_{I\Pi}(1|10)$, we have
$H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

####  2.5.2. $p=3$.

######  Lemma.

(a) For ${\mathfrak{g}}={\mathfrak{gl}}(6)$, ${\mathfrak{gl}}(4|2)$, and
${\mathfrak{gl}}(3|3)$, the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is
spanned by the following cocycle (here $h_{6}$ is the outer derivation; same
situation as for ${\mathfrak{gl}}(3)$ for $p=3$):

$c_{0}=2\ h_{1}\otimes dh_{6}+h_{2}\otimes dh_{6}+2\ h_{4}\otimes
dh_{6}+h_{5}\otimes dh_{6}$

(b) For ${\mathfrak{g}}={\mathfrak{psl}}(6)$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycle
turning ${\mathfrak{psl}}$ into ${\mathfrak{pgl}}$:

$\begin{array}[]{ll}c_{0}=&2x_{3}\otimes dx_{3}+x_{4}\otimes
dx_{4}+2x_{7}\otimes dx_{7}+x_{9}\otimes dx_{9}+2x_{10}\otimes
dx_{10}+y_{3}\otimes dy_{3}\\\ &+2y_{4}\otimes dy_{4}+y_{7}\otimes
dy_{7}+2y_{9}\otimes dy_{9}+y_{10}\otimes dy_{10}\end{array}$

(c) For ${\mathfrak{g}}={\mathfrak{psl}}(3|3)$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycle
turning ${\mathfrak{psl}}$ into ${\mathfrak{pgl}}$:

$\begin{array}[]{ll}c_{0}=&2x_{3}\otimes dx_{3}+2x_{4}\otimes
dx_{4}+2x_{7}\otimes dx_{7}+2x_{9}\otimes dx_{9}+2x_{10}\otimes
dx_{10}+y_{3}\otimes dy_{3}\\\ &+y_{4}\otimes dy_{4}+y_{7}\otimes
dy_{7}+y_{9}\otimes dy_{9}+y_{10}\otimes dy_{10}\end{array}$

######  Lemma.

a) For ${\mathfrak{g}}={\mathfrak{g}}(8,3),{\mathfrak{g}}=\mathfrak{el}(5;3)$,
we have $H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

b) For ${\mathfrak{g}}={\mathfrak{g}}(2,6)$ with Cartan matrix and basis

$\begin{pmatrix}2&-1&0&0&0\\\ -1&2&-1&0&0\\\ 0&-1&2&-1&-1\\\ 0&0&-1&2&0\\\
0&0&1&0&0\end{pmatrix}\ \ \footnotesize{\begin{array}[]{l}x_{1},\ x_{2},\
x_{3},\ x_{4},\ x_{5},\\\ x_{6}=[x_{1},x_{2}],\ x_{7}=[x_{2},x_{3}],\
x_{8}=[x_{3},x_{4}],\ x_{9}=[x_{3},x_{5}],\\\ x_{10}=[x_{3},[x_{1},x_{2}]],\
x_{11}=[x_{4},[x_{2},x_{3}]],\ x_{12}=[x_{5},[x_{2},x_{3}]],\
x_{13}=[x_{5},[x_{3},x_{4}]],\\\ x_{14}=[x_{5},[x_{4},[x_{2},x_{3}]]],\
x_{15}=[[x_{1},x_{2}],[x_{3},x_{4}]],\
x_{16}=[[x_{1},x_{2}],[x_{3},x_{5}]],\\\
x_{17}=[[x_{1},x_{2}],[x_{5},[x_{3},x_{4}]]],\
x_{18}=[[x_{3},x_{5}],[x_{4},[x_{2},x_{3}]]],\\\
x_{19}=[[x_{3},[x_{1},x_{2}]],[x_{5},[x_{3},x_{4}]]],\
x_{20}=[[x_{5},[x_{2},x_{3}]],[x_{5},[x_{3},x_{4}]]],\\\
x_{21}=[[x_{5},[x_{2},x_{3}]],[[x_{1},x_{2}],[x_{3},x_{4}]]],\
[x_{22}=[[x_{5},[x_{3},x_{4}]],[[x_{1},x_{2}],[x_{3},x_{5}]]],\\\
x_{23}=[[x_{5},[x_{4},[x_{2},x_{3}]]],[[x_{1},x_{2}],[x_{3},x_{5}]]],\\\
x_{24}=[[[x_{1},x_{2}],[x_{3},x_{5}]],[[x_{3},x_{5}],[x_{4},[x_{2},x_{3}]]]],\\\
x_{25}=[[[x_{1},x_{2}],[x_{5},[x_{3},x_{4}]]],[[x_{3},x_{5}],[x_{4},[x_{2},x_{3}]]]]\end{array}}$

and grading operator $h_{6}=(1,0,0,0,0)$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycle

$c_{0}=2\ h_{1}\otimes dh_{6}+h_{2}\otimes dh_{6}+h_{5}\otimes dh_{6}$

For ${\mathfrak{g}}(2,6)^{(1)}/{\mathfrak{c}}$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the cocycle

$\footnotesize{\begin{array}[]{ll}c_{0}=&2\ x_{4}\otimes dx_{4}+2\
x_{5}\otimes dx_{5}+2\ x_{8}\otimes dx_{8}+2\ x_{9}\otimes dx_{9}+2\
x_{11}\otimes dx_{11}+2\ x_{12}\otimes dx_{12}+2\ x_{15}\otimes dx_{15}+\\\
&2\ x_{16}\otimes dx_{16}+x_{20}\otimes dx_{20}+x_{22}\otimes
dx_{22}+x_{23}\otimes dx_{23}+x_{24}\otimes dx_{24}+y_{4}\otimes
dy_{4}+y_{5}\otimes dy_{5}+y_{8}\otimes dy_{8}+\\\ &y_{9}\otimes
dy_{9}+y_{11}\otimes dy_{11}+y_{12}\otimes dy_{12}+y_{15}\otimes
dy_{15}+y_{16}\otimes dy_{16}+2\ y_{20}\otimes dy_{20}+\\\ &2\ y_{22}\otimes
dy_{22}+2\ y_{23}\otimes dy_{23}+2\ y_{24}\otimes dy_{24}\end{array}}$

####  2.5.3. $p=5$.

######  Lemma.

(a) For ${\mathfrak{g}}={\mathfrak{gl}}(3|3)$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycle
(here $h_{6}$ is the outer derivation; same situation as for
${\mathfrak{gl}}(3)$ for $p=3$):

$c_{0}=4\ h_{1}\otimes dh_{6}+3\ h_{2}\otimes dh_{6}+2\ h_{3}\otimes dh_{6}+2\
h_{4}\otimes dh_{6}+h_{5}\otimes dh_{6}$

(b) For ${\mathfrak{g}}={\mathfrak{psl}}(3|3)$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycle
turning ${\mathfrak{psl}}$ into ${\mathfrak{pgl}}$:

$\begin{array}[]{ll}c_{0}=&4\ x_{2}\otimes dx_{2}+4\ x_{3}\otimes dx_{3}+4\
x_{6}\otimes dx_{6}+4\ x_{8}\otimes dx_{8}+4\ x_{12}\otimes
dx_{12}+y_{2}\otimes dy_{2}\\\ &+y_{3}\otimes dy_{3}+y_{6}\otimes
dy_{6}+y_{8}\otimes dy_{8}+y_{12}\otimes dy_{12}\end{array}$

######  Lemma.

For ${\mathfrak{g}}=\mathfrak{el}(5;5)$, we have
$H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

###  2.6. Rank=6

####  2.6.1. $p=2$.

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{e}}(6;1)$, ${\mathfrak{e}}(6,6)$, we have
$H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

####  2.6.2. $p=3$.

######  Lemma.

For ${\mathfrak{g}}=\mathfrak{g}(4;8)$, $\mathfrak{g}(6;6)$, we have
$H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

###  2.7. Rank=7

####  2.7.1. $p=2$.

######  Lemma.

Let the Chevalley basis of ${\mathfrak{g}}={\mathfrak{e}}(7)$ (those of
${\mathfrak{g}}={\mathfrak{e}}(7,i)$ differ in that
$p(x_{i})=p(y_{i})={\bar{1}}$, the other Chevalley generators being even) be

${\tiny\begin{array}[]{l}x_{1},\ x_{2},\ x_{3},\ x_{4},\ x_{5},\ x_{6},\
x_{7},\\\ x_{8}=[x_{1},x_{2}],\ x_{9}=[x_{2},x_{3}],\ x_{10}=[x_{3},x_{4}],\
x_{11}=[x_{4},x_{5}],\ x_{12}=[x_{4},x_{7}],\ x_{13}=[x_{5},x_{6}],\\\
x_{14}=[x_{3},[x_{1},x_{2}]],\ x_{15}=[x_{4},[x_{2},x_{3}]],\
x_{16}=[x_{5},[x_{3},x_{4}]],\ x_{17}=[x_{6},[x_{4},x_{5}]],\\\ \hskip
14.22636ptx_{18}=[x_{7},[x_{3},x_{4}]],\ x_{19}=[x_{7},[x_{4},x_{5}]],\\\
x_{20}=[x_{7},[x_{5},[x_{3},x_{4}]]],\ x_{21}=[[x_{1},x_{2}],[x_{3},x_{4}]],\
x_{22}=[[x_{2},x_{3}],[x_{4},x_{5}]],\\\ \hskip
14.22636ptx_{23}=[[x_{2},x_{3}],[x_{4},x_{7}]],\
x_{24}=[[x_{3},x_{4}],[x_{5},x_{6}]],\
x_{25}=[[x_{4},x_{7}],[x_{5},x_{6}]],\\\
x_{26}=[[x_{2},x_{3}],[x_{7},[x_{4},x_{5}]]],\
x_{27}=[[x_{4},x_{5}],[x_{3},[x_{1},x_{2}]]],\
x_{28}=[[x_{4},x_{7}],[x_{3},[x_{1},x_{2}]]],\\\ \hskip
14.22636ptx_{29}=[[x_{4},x_{7}],[x_{5},[x_{3},x_{4}]]],\
x_{30}=[[x_{5},x_{6}],[x_{4},[x_{2},x_{3}]]],\
x_{31}=[[x_{5},x_{6}],[x_{7},[x_{3},x_{4}]]],\\\
x_{32}=[[x_{5},x_{6}],[[x_{2},x_{3}],[x_{4},x_{7}]]],\
x_{33}=[[x_{3},[x_{1},x_{2}]],[x_{6},[x_{4},x_{5}]]],\
x_{34}=[[x_{3},[x_{1},x_{2}]],[x_{7},[x_{4},x_{5}]]],\\\ \hskip
14.22636ptx_{35}=[[x_{4},[x_{2},x_{3}]],[x_{7},[x_{4},x_{5}]]],\
x_{36}=[[x_{6},[x_{4},x_{5}]],[x_{7},[x_{3},x_{4}]]],\\\
x_{37}=[[x_{3},[x_{1},x_{2}]],[[x_{4},x_{7}],[x_{5},x_{6}]]],\
x_{38}=[[x_{6},[x_{4},x_{5}]],[[x_{2},x_{3}],[x_{4},x_{7}]]],\
x_{39}=[[x_{7},[x_{3},x_{4}]],[[x_{2},x_{3}],[x_{4},x_{5}]]],\\\ \hskip
14.22636ptx_{40}=[[x_{7},[x_{4},x_{5}]],[[x_{1},x_{2}],[x_{3},x_{4}]]],\
x_{41}=[[x_{7},[x_{4},x_{5}]],[[x_{3},x_{4}],[x_{5},x_{6}]]],\\\
x_{42}=[[x_{7},[x_{5},[x_{3},x_{4}]]],[[x_{1},x_{2}],[x_{3},x_{4}]]],\
x_{43}=[[[x_{1},x_{2}],[x_{3},x_{4}]],[[x_{4},x_{7}],[x_{5},x_{6}]]],\\\
\hskip
14.22636ptx_{44}=[[[x_{2},x_{3}],[x_{4},x_{5}]],[[x_{4},x_{7}],[x_{5},x_{6}]]],\
x_{45}=[[[x_{2},x_{3}],[x_{4},x_{7}]],[[x_{3},x_{4}],[x_{5},x_{6}]]],\\\
x_{46}=[[[x_{2},x_{3}],[x_{4},x_{7}]],[[x_{4},x_{5}],[x_{3},[x_{1},x_{2}]]]],\
x_{47}=[[[x_{3},x_{4}],[x_{5},x_{6}]],[[x_{2},x_{3}],[x_{7},[x_{4},x_{5}]]]],\\\
\hskip
14.22636ptx_{48}=[[[x_{3},x_{4}],[x_{5},x_{6}]],[[x_{4},x_{7}],[x_{3},[x_{1},x_{2}]]]],\
x_{49}=[[[x_{4},x_{7}],[x_{5},x_{6}]],[[x_{4},x_{5}],[x_{3},[x_{1},x_{2}]]]],\\\
x_{50}=[[[x_{4},x_{5}],[x_{3},[x_{1},x_{2}]]],[[x_{5},x_{6}],[x_{7},[x_{3},x_{4}]]]],\
x_{51}=[[[x_{4},x_{7}],[x_{3},[x_{1},x_{2}]]],[[x_{5},x_{6}],[x_{4},[x_{2},x_{3}]]]],\\\
x_{52}=[[[x_{4},x_{7}],[x_{5},[x_{3},x_{4}]]],[[x_{5},x_{6}],[x_{4},[x_{2},x_{3}]]]],\\\
x_{53}=[[[x_{4},x_{7}],[x_{5},[x_{3},x_{4}]]],[[x_{3},[x_{1},x_{2}]],[x_{6},[x_{4},x_{5}]]]],\
x_{54}=[[[x_{5},x_{6}],[x_{4},[x_{2},x_{3}]]],[[x_{3},[x_{1},x_{2}]],[x_{7},[x_{4},x_{5}]]]],\\\
\hskip
14.22636ptx_{55}=[[[x_{5},x_{6}],[x_{7},[x_{3},x_{4}]]],[[x_{4},[x_{2},x_{3}]],[x_{7},[x_{4},x_{5}]]]],\\\
x_{56}=[[[x_{3},[x_{1},x_{2}]],[x_{6},[x_{4},x_{5}]]],[[x_{4},[x_{2},x_{3}]],[x_{7},[x_{4},x_{5}]]]],\\\
\hskip
14.22636ptx_{57}=[[[x_{3},[x_{1},x_{2}]],[x_{7},[x_{4},x_{5}]]],[[x_{6},[x_{4},x_{5}]],[x_{7},[x_{3},x_{4}]]]],\\\
x_{58}=[[[x_{3},[x_{1},x_{2}]],[x_{6},[x_{4},x_{5}]]],[[x_{7},[x_{3},x_{4}]],[[x_{2},x_{3}],[x_{4},x_{5}]]]],\\\
\hskip
14.22636ptx_{59}=[[[x_{4},[x_{2},x_{3}]],[x_{7},[x_{4},x_{5}]]],[[x_{3},[x_{1},x_{2}]],[[x_{4},x_{7}],[x_{5},x_{6}]]]],\\\
x_{60}=[[[x_{3},[x_{1},x_{2}]],[[x_{4},x_{7}],[x_{5},x_{6}]]],[[x_{7},[x_{3},x_{4}]],[[x_{2},x_{3}],[x_{4},x_{5}]]]],\\\
x_{61}=[[[x_{7},[x_{4},x_{5}]],[[x_{1},x_{2}],[x_{3},x_{4}]]],[[[x_{2},x_{3}],[x_{4},x_{7}]],[[x_{3},x_{4}],[x_{5},x_{6}]]]],\\\
x_{62}=[[[x_{7},[x_{5},[x_{3},x_{4}]]],[[x_{1},x_{2}],[x_{3},x_{4}]]],[[[x_{2},x_{3}],[x_{4},x_{5}]],[[x_{4},x_{7}],[x_{5},x_{6}]]]],\\\
x_{63}=[[[[x_{2},x_{3}],[x_{4},x_{7}]],[[x_{3},x_{4}],[x_{5},x_{6}]]],[[[x_{4},x_{7}],[x_{5},x_{6}]],[[x_{4},x_{5}],[x_{3},[x_{1},x_{2}]]]]]\end{array}}$

(a) For ${\mathfrak{g}}={\mathfrak{e}}(7,i)$, where $i=1,6$ or $7$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycle:

$c_{0}=h_{1}\otimes dh_{8}+h_{3}\otimes dh_{8}+h_{7}\otimes dh_{8}$

(b) For ${\mathfrak{g}}={\mathfrak{e}}^{(1)}(7,i)/{\mathfrak{c}}$, where
$i=1,6$ or $7$, the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by
the following cocycle

$\footnotesize{\begin{array}[]{ll}c_{0}=&x_{2}\otimes dx_{2}+x_{4}\otimes
dx_{4}+x_{7}\otimes dx_{7}+x_{8}\otimes dx_{8}+x_{9}\otimes
dx_{9}+x_{10}\otimes dx_{10}+x_{11}\otimes dx_{11}+x_{14}\otimes dx_{14}\\\
&+x_{16}\otimes dx_{16}+x_{17}\otimes dx_{17}+x_{23}\otimes
dx_{23}+x_{24}\otimes dx_{24}+x_{26}\otimes dx_{26}+x_{28}\otimes
dx_{28}+x_{29}\otimes dx_{29}\\\ &+x_{32}\otimes dx_{32}+x_{34}\otimes
dx_{34}+x_{36}\otimes dx_{36}+x_{37}\otimes dx_{37}+x_{41}\otimes
dx_{41}+x_{46}\otimes dx_{46}+x_{51}\otimes dx_{51}\\\ &+x_{52}\otimes
dx_{52}+x_{53}\otimes dx_{53}+x_{54}\otimes dx_{54}+x_{59}\otimes
dx_{59}+x_{60}\otimes dx_{60}+y_{2}\otimes dy_{2}+y_{4}\otimes dy_{4}\\\
&+y_{7}\otimes dy_{7}+y_{8}\otimes dy_{8}+y_{9}\otimes dy_{9}+y_{10}\otimes
dy_{10}+y_{11}\otimes dy_{11}+y_{14}\otimes dy_{14}+y_{16}\otimes dy_{16}\\\
&+y_{17}\otimes dy_{17}+y_{23}\otimes dy_{23}+y_{24}\otimes
dy_{24}+y_{26}\otimes dy_{26}+y_{28}\otimes dy_{28}+y_{29}\otimes
dy_{29}+y_{32}\otimes dy_{32}\\\ &+y_{34}\otimes dy_{34}+y_{36}\otimes
dy_{36}+y_{37}\otimes dy_{37}+y_{41}\otimes dy_{41}+y_{46}\otimes
dy_{46}+y_{51}\otimes dy_{51}+y_{52}\otimes dy_{52}\\\ &+y_{53}\otimes
dy_{53}+y_{54}\otimes dy_{54}+y_{59}\otimes dy_{59}+y_{60}\otimes
dy_{60}\end{array}}$

####  2.7.2. $p=3$.

######  Lemma.

For ${\mathfrak{g}}=\mathfrak{g}(8;8)$, we have
$H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

####  2.7.3. Rank=8.

####  2.7.4. $p=2$.

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{e}}(8),{\mathfrak{e}}(8;1)$,
${\mathfrak{e}}(8,8)$, we have $H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

####  2.7.5. $p>2$.

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{e}}(8)$, we have
$H^{1}({\mathfrak{g}};{\mathfrak{g}})=0$.

###  2.8. Symmetric queerifications

######  Lemma.

For ${\mathfrak{g}}=\mathfrak{q}^{(1)}({\mathfrak{wk}}(3;a)/{\mathfrak{c}})$,
see (9), the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the
following three cocycles:

$\begin{array}[]{ll}c_{0}^{1}=&h_{1}\otimes d(\Pi(h_{1}))+h_{2}\otimes
d(\Pi(h_{2}))+x_{1}\otimes d(\Pi(x_{1}))+x_{5}\otimes
d(\Pi(x_{5}))+x_{6}\otimes d(\Pi(x_{6}))+y_{1}\otimes d(\Pi(y_{1}))+\\\
&y_{5}\otimes d(\Pi(y_{5}))+y_{6}\otimes d(\Pi(y_{6}))+\Pi(x_{2})\otimes
dx_{2}+\Pi(x_{3})\otimes dx_{3}+\Pi(x_{4})\otimes d(x_{4})+\Pi(x_{7})\otimes
dx_{7}+\\\ &\Pi(y_{2})\otimes dy_{2}+\Pi(y_{3})\otimes
dy_{3}+\Pi(y_{4})\otimes dy_{4}+\Pi(y_{7})\otimes d(y_{7})\\\\[8.53581pt]
c_{0}^{2}=&x_{3}\otimes dx_{3}+x_{5}\otimes x_{5}+x_{6}\otimes
dx_{6}+x_{7}\otimes dx_{7}+y_{3}\otimes dy_{3}+y_{5}\otimes
dy_{5}+y_{6}\otimes dy_{6}+y_{7}\otimes d(y_{7})+\\\ &\Pi(h_{1})\otimes
d(\Pi(h_{1}))+\Pi(h_{2})\otimes d(\Pi(h_{2}))+\Pi(x_{1})\otimes
d(\Pi(x_{1}))+\Pi(x_{2})\otimes d(\Pi(x_{2}))+\Pi(x_{4})\otimes
d(\Pi(x_{4}))+\\\ &\Pi(y_{1})\otimes d(\Pi(y_{1}))+\Pi(y_{2})\otimes
d(\Pi(y_{2}))+\Pi(y_{4})\otimes d(\Pi(y_{4}))\\\\[8.53581pt]
c_{0}^{3}=&h_{1}\otimes d(\Pi(h_{1}))+h_{2}\otimes d(\Pi(h_{2}))+x_{1}\otimes
d(\Pi(x_{1}))+x_{2}\otimes d(\Pi(x_{2}))+x_{3}\otimes
d(\Pi(x_{3}))+x_{4}\otimes d(\Pi(x_{4}))+\\\ &x_{5}\otimes
d(\Pi(x_{5}))+x_{6}\otimes d(\Pi(x_{6}))+x_{7}\otimes
d(\Pi(x_{7}))+y_{1}\otimes d(\Pi(y_{1}))+y_{2}\otimes
d(\Pi(y_{2}))+y_{3}\otimes d(\Pi(y_{3}))+\\\ &y_{4}\otimes
d(\Pi(y_{4}))+y_{5}\otimes d(\Pi(y_{5}))+y_{6}\otimes
d(\Pi(y_{6}))+y_{7}\otimes d(\Pi(y_{7}))\end{array}$

######  Lemma.

For ${\mathfrak{g}}=\mathfrak{q}({\mathfrak{wk}}(4;a))$ for the following
Cartan matrix and basis of ${\mathfrak{wk}}(4;a)$

(18) $\begin{pmatrix}0&\alpha&1&0\\\ \alpha&0&0&0\\\ 1&0&0&1\\\
0&0&1&0\end{pmatrix}\quad\begin{array}[]{l}x_{1},\ x_{2},\ x_{3},\ x_{4},\\\
x_{5}=[x_{1},x_{2}],\ x_{6}=[x_{1},x_{3}],\ x_{7}=[x_{3},x_{4}],\\\
x_{8}=[x_{3},[x_{1},x_{2}]],\ x_{9}=[x_{4},[x_{1},x_{3}]],\\\
x_{10}=[[x_{1},x_{2}],[x_{1},x_{3}]],\
x_{11}=[[x_{1},x_{2}],[x_{3},x_{4}]],\\\
x_{12}=[[x_{1},x_{2}],[x_{4},[x_{1},x_{3}]]],\\\
x_{13}=[[x_{3},[x_{1},x_{2}]],[x_{4},[x_{1},x_{3}]]],\\\
x_{14}=[[x_{4},[x_{1},x_{3}]],[[x_{1},x_{2}],[x_{1},x_{3}]]],\\\
x_{15}=[[[x_{1},x_{2}],[x_{1},x_{3}]],[[x_{1},x_{2}],[x_{3},x_{4}]]]\end{array}$

the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
two cocycles:

$\begin{array}[]{ll}c_{0}^{1}=&h_{1}\otimes d(\Pi(y_{1}))+h_{2}\otimes
d(\Pi(h_{2}))+h_{3}\otimes d(\Pi(h_{3})+h_{4}\otimes
d(\Pi(h_{4}))+x_{1}\otimes d(\Pi(x_{1}))+x_{2}\otimes d(\Pi(x_{2}))+\\\
&x_{3}\otimes d(\Pi(x_{3}))+x_{4}\otimes d(\Pi(x_{4}))+x_{5}\otimes
d(\Pi(x_{5}))+x_{6}\otimes d(\Pi(x_{6}))+x_{7}\otimes
d(\Pi(x_{7}))+x_{8}\otimes d(\Pi(x_{8}))+\\\ &x_{9}\otimes
d(\Pi(x_{9}))+x_{10}\otimes d(\Pi(x_{10}))+x_{11}\otimes
d(\Pi(x_{11}))+x_{12}\otimes d(\Pi(x_{12}))+x_{13}\otimes d(\Pi(x_{13}))+\\\
&x_{14}\otimes d(\Pi(x_{14}))+x_{15}\otimes d(\Pi(x_{15})+y_{1}\otimes
d(\Pi(y_{1}))+y_{2}\otimes d(\Pi(y_{2}))+y_{3}\otimes
d(\Pi(y_{3}))+y_{4}\otimes d(\Pi(y_{4}))+\\\ &y_{5}\otimes
d(\Pi(y_{5}))+y_{6}\otimes d(\Pi(y_{6}))+y_{7}\otimes
d(\Pi(y_{7}))+y_{8}\otimes d(\Pi(y_{8}))+y_{9}\otimes
d(\Pi(y_{9}))+y_{10}\otimes d(\Pi(y_{10}))+\\\ &y_{11}\otimes
d(\Pi(y_{11}))+y_{12}\otimes d(\Pi(y_{12}))+y_{13}\otimes
d(\Pi(y_{13}))+y_{14}\otimes d(\Pi(y_{14}))+y_{15}\otimes
d(\Pi(y_{15}))\\\\[8.53581pt] c_{0}^{2}=&x_{4}\otimes dx_{4}+x_{7}\otimes
dx_{7}+x_{9}\otimes dx_{9}+x_{11}\otimes dx_{11}+x_{12}\otimes
dx_{12}+x_{13}\otimes dx_{13}+x_{14}\otimes dx_{14}+x_{15}\otimes dx_{15}+\\\
&y_{4}\otimes dy_{4}+y_{7}\otimes dy_{7}+y_{9}\otimes dy_{9}+y_{11}\otimes
dy_{11}+y_{12}\otimes dy_{12}+y_{13}\otimes dy_{13}+y_{14}\otimes
dy_{14}+y_{15}\otimes dy_{15}+\\\ &\Pi(h_{1})\otimes
d(\Pi(h_{1}))+\Pi(h_{2})\otimes d(\Pi(h_{2}))+\Pi(h_{3})\otimes
d(\Pi(h_{3})+\Pi(h_{4})\otimes d(\Pi(h_{4}))+\Pi(x_{1})\otimes
d(\Pi(x_{1}))+\\\ &\Pi(x_{2})\otimes d(\Pi(x_{2}))+\Pi(x_{3})\otimes
d(\Pi(x_{3}))+\Pi(x_{5})\otimes d(\Pi(x_{5}))+\Pi(x_{6})\otimes
d(\Pi(x_{6}))+\Pi(x_{8})\otimes d(\Pi(x_{8}))+\\\ &\Pi(x_{10})\otimes
d(\Pi(x_{10}))+\Pi(y_{1})\otimes d(\Pi(y_{1}))+\Pi(y_{2})\otimes
d(\Pi(y_{2}))+\Pi(y_{3})\otimes d(\Pi(y_{3}))+\Pi(y_{5})\otimes
d(\Pi(y_{5}))+\\\ &\Pi(y_{6})\otimes d(\Pi(y_{6}))+\Pi(y_{8})\otimes
d(\Pi(y_{8}))+\Pi(y_{10})\otimes d(\Pi(y_{10}))\end{array}$

###  2.9. Orthogonal algebras of the type ${\mathfrak{o}}_{I}$ and their
derived

Let $X_{i,j}:=E^{i,j}+E^{j,i}$.

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{o}}^{(1)}_{I}(n)$, where $n\neq 4$. the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by $n-1$ cocycles of degree
$0$ and the following form:

$\begin{array}[]{l}X_{1,2}\otimes dX_{1,2}+X_{1,3}\otimes
dX_{1,3}+\dots+X_{1,n}\otimes dX_{1,n},\\\ X_{1,2}\otimes
dX_{1,2}+X_{2,3}\otimes dX_{2,3}+\dots+X_{2,n}\otimes dX_{2,n},\\\
X_{1,3}\otimes dX_{1,3}+X_{2,3}\otimes dX_{2,3}+\dots+X_{3,n}\otimes
dX_{3,n},\\\ \dotfill\\\ X_{1,n-1}\otimes dX_{1,n-1}+X_{2,n-1}\otimes
dX_{2,n-1}+\dots+X_{n-1,n}\otimes dX_{n-1,n}.\end{array}$

######  Lemma.

For ${\mathfrak{g}}={\mathfrak{o}}^{(1)}_{I}(4)$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following cocycles of
degree $0$:

$\begin{array}[]{ll}\text{regular cases}&\begin{cases}X_{1,2}\otimes
dX_{1,2}+X_{1,3}\otimes dX_{1,3}+X_{1,4}\otimes dX_{1,4}\\\ X_{1,2}\otimes
dX_{1,2}+X_{2,3}\otimes dX_{2,3}+X_{2,4}\otimes dX_{2,4}\\\ X_{1,3}\otimes
dX_{1,3}+X_{2,3}\otimes dX_{2,3}+X_{3,4}\otimes dX_{3,4}\\\\[5.69054pt]
\end{cases}\\\ \text{exceptional cases}&\begin{cases}X_{1,2}\otimes
dX_{3,4}+X_{1,3}\otimes dX_{2,4}+X_{2,3}\otimes dX_{1,4}\\\ X_{1,2}\otimes
dX_{3,4}+X_{1,4}\otimes dX_{2,3}+X_{2,4}\otimes dX_{1,3}\\\ X_{1,3}\otimes
dX_{2,4}+X_{1,4}\otimes dX_{2,3}+X_{3,4}\otimes dX_{1,2}\\\
\end{cases}\end{array}$

##  3\. Periplectic Lie superalgebras and their desuperizations

###  3.1. The Lie superalgebras preserving symmetric odd bilinear forms for
$p>2$

We consider the following three types of periplectic forms, i.e., symmetric
odd bilinear forms with the Gram matrix $\Pi_{n|n}$:

1) The Lie superalgebra ${\mathfrak{pe}}(n)$ that preserves this form,

2) ${\mathfrak{spe}}(n):={\mathfrak{pe}}(n)\cap{\mathfrak{sle}}(n|n)$,

3) ${\mathfrak{p}}{\mathfrak{spe}}(pn):={\mathfrak{spe}}(pn)/{\mathfrak{c}}$.

These three types of algebras, as well as the Lie superalgebras of their outer
derivations and central extensions are ingredients needed for the input of the
generalized Cartan prolongations and possible new simple Lie superalgebras
(analogs of Lie algebras of Hamiltonian vector fields) are concerned.

Recall that for $p=0$, ${\mathfrak{pe}}(n)$ has neither non-trivial extensions
nor outer derivatives; ${\mathfrak{spe}}(n)$ has only a central extension for
$n=4$, and the derivative $d=\begin{pmatrix}1_{n}&0\\\ 0&-1_{n}\end{pmatrix}$
for any $n>2$.

######  Lemma.

For $p>2$, we have $H^{2}({\mathfrak{pe}}(n))=0$. Similarly,
$H^{2}({\mathfrak{spe}}(n))=0$ for $n>4$ and $p>3$. Moreover,
$H^{2}({\mathfrak{spe}}(3))=0$ for $p>3$; however, for $p=3$ it is spanned by
the following three cocycles

$\begin{array}[]{ccl}c_{0}^{1}&=&d(E^{2,3}-E^{6,5})\wedge
d(E^{1,3}-E^{6,4})+d(E^{1,5}-E^{2,4})\wedge d(E^{6,3})\\\\[8.53581pt]
c_{0}^{2}&=&d(E^{2,3}-E^{5,6})\wedge
d(E^{1,2}-E^{5,4})+d(E^{1,6}-E^{3,4})\wedge d(E^{5,2})\\\\[8.53581pt]
c_{0}^{3}&=&2d(E^{2,1}-E^{4,5})\wedge
d(E^{3,1}-E^{4,6})+d(E^{2,6}-E^{3,5})\wedge d(E^{4,1})\end{array}$

In addition, for every $p>2$, the space $H^{2}({\mathfrak{spe}}(4))$ is
spanned by the cocycle

$c_{-2}=d(E^{1,6}-E^{2,5})\wedge d(E^{3,8}-E^{4,7})-d(E^{1,7}-E^{3,5})\wedge
d(E^{2,8}-E^{4,6})+d(E^{1,8}-E^{4,5})\wedge d(E^{2,7}-E^{3,8}).$

######  Lemma.

For $p>2$, we have $H^{1}({\mathfrak{pe}}(n);{\mathfrak{pe}}(n))=0$. Moreover,
the space $H^{1}({\mathfrak{spe}}(n);{\mathfrak{spe}}(n))$ is spanned by the
following derivative $d=\begin{pmatrix}1_{n}&0\\\ 0&-1_{n}\end{pmatrix}$ for
any $n>2$.

###  3.2. The Lie superalgebras preserving symmetric odd bilinear forms for
$p=2$ and their desuperizations

For $p=2$, the analogs of symplectic and periplectic Lie (super)algebras
accrue additional elements: If the matrix of the bilinear form ${\mathcal{B}}$
is $\Pi_{2n}$ (resp. $\Pi_{n|n}$), then ${\mathfrak{aut}}({\mathcal{B}})$
consists of the (super)matrices of the form

(19) $\begin{pmatrix}A&B\\\ C&A^{t}\end{pmatrix},$

where $B$ and $C$ are symmetric. Denote these general Lie (super)algebras
${\mathfrak{aut}}_{gen}({\mathcal{B}})$; in particular, for
${\mathcal{B}}=\Pi_{2n}$ (resp. $\Pi_{n|n}$) we write
${\mathfrak{o}}_{gen}(2n)$ (resp. ${\mathfrak{pe}}_{gen}(n)$). Clearly,
${\mathfrak{o}}_{gen}(2n)\simeq{\mathfrak{pe}}_{gen}(n)$ as spaces.

Let $ZD$ denote the space of symmetric matrices with zeros on their main
diagonals. The derived Lie (super)algebra
${\mathfrak{aut}}^{(1)}({\mathcal{B}})$ consists of the (super)matrices of the
form (19), where $B,C\in ZD$. In other words, these Lie (super)algebras
resemble the orthogonal Lie algebras. On these Lie (super)algebras
${\mathfrak{aut}}^{(1)}({\mathcal{B}})$ the following (super)trace — we call
it the _half-trace_ — is defined:

(20) ${\text{htr}}:\begin{pmatrix}A&B\\\
C&A^{t}\end{pmatrix}\longrightarrow\mathop{\mathrm{tr}}\nolimits A.$

The half-traceless Lie sub(super)algebra of
${\mathfrak{aut}}^{(1)}({\mathcal{B}})$ is isomorphic to
${\mathfrak{aut}}^{(2)}({\mathcal{B}})$.

There is, however, a Lie superalgebra
$\widetilde{\mathfrak{aut}}({\mathcal{B}})$, such that
${\mathfrak{aut}}^{(1)}({\mathcal{B}})\subset\widetilde{\mathfrak{aut}}({\mathcal{B}})\subset{\mathfrak{aut}}({\mathcal{B}})$,
consisting of (super)matrices of the form (19), where $C\in ZD$ while $B$ is
just symmetric (obviously, there is an isomorphic version of
$\widetilde{\mathfrak{aut}}({\mathcal{B}})$ in which $B\in ZD$ and $C$ is just
symmetric). Denote this $\widetilde{\mathfrak{aut}}({\mathcal{B}})$ by
${\mathfrak{o}}{\mathfrak{p}}(2n)$ if ${\mathcal{B}}$ is even and
${\mathfrak{pe}}(n)$ if ${\mathcal{B}}$ is odd.

We have, therefore, the following cases to consider assuming, as in p.4, $\deg
A^{i,i+1}=-\deg A^{i+1,i}=1$, and setting the degree of the LOWEST (resp.
HIGHEST) weight vector in the ${\mathfrak{gl}}(n)$-module of matrices $B$
(resp. $C$) in (19) be equal to 1 (resp. $-1$):

1) ${\mathfrak{o}}_{gen}(2n)$ and ${\mathfrak{pe}}_{gen}(n)$ (in (19), both
$B$ and $C$ symmetric);

1a)
${\mathfrak{p}}{\mathfrak{o}}_{gen}(2n):={\mathfrak{o}}_{gen}(2n)/{\mathbb{K}}\
1_{2n}$ and
${\mathfrak{p}}{\mathfrak{pe}}_{gen}(n):={\mathfrak{pe}}_{gen}(n)/{\mathbb{K}}\
1_{2n}$;

2) ${\mathfrak{o}}{\mathfrak{p}}(2n)$ and ${\mathfrak{pe}}(n)$ (in (19), both
$B$ and $C$ symmetric, but $C\in ZD$);

2a)
${\mathfrak{p}}{\mathfrak{o}}{\mathfrak{p}}(2n):={\mathfrak{o}}_{gen}(2n)/{\mathbb{K}}\
1_{2n}$ and
${\mathfrak{p}}{\mathfrak{pe}}(n):={\mathfrak{pe}}_{gen}(n)/{\mathbb{K}}\
1_{2n}$;

2b)${\mathfrak{o}}^{(1)}(2n)$ (in (19), both $B\in ZD$ and $C\in ZD$);

2c)${\mathfrak{p}}{\mathfrak{o}}^{(1)}(2n):={\mathfrak{o}}^{(1)}(2n/{\mathbb{K}}\
1_{2n}$;

3) ${\mathfrak{s}}{\mathfrak{o}}{\mathfrak{p}}(2n)$ and ${\mathfrak{spe}}(n)$
(in (19), both $B$ and $C$ symmetric, but $C\in ZD$, and
$\mathop{\mathrm{tr}}\nolimits A=0$);

3a)
${\mathfrak{p}}{\mathfrak{s}}{\mathfrak{o}}{\mathfrak{p}}(2n):={\mathfrak{s}}{\mathfrak{o}}{\mathfrak{p}}(2n)/{\mathbb{K}}\
1_{2n}$ and
${\mathfrak{p}}{\mathfrak{spe}}(n):={\mathfrak{spe}}(n)/{\mathbb{K}}\ 1_{2n}$;

4) ${\mathfrak{o}}^{(2)}(2n)$ (in (19), both $B\in ZD$ and $C\in ZD$, and
$\mathop{\mathrm{tr}}\nolimits A=0$);

4a)
${\mathfrak{p}}{\mathfrak{o}}^{(2)}(2n):={\mathfrak{o}}^{(2)}(2n)/{\mathbb{K}}\
1_{2n}$.

######  Lemma.

For $p=2$, the space $H^{2}({\mathfrak{o}}_{gen}(2n))$ is spanned by the
following cocycles

(21) $\begin{array}[]{ll}c_{-2}^{i,j}=d(E^{n+i,i})\wedge d(E^{n+j,j}),&1\leq
i<j\leq n\\\\[8.53581pt] c_{-1}^{j}=d(E^{j,j}+E^{n+j,n+j})\wedge
d(E^{n+j,j})+&\\\ \hskip
56.9055pt\displaystyle\sum_{i\not=j}d(E^{i,j}+E^{j+n,i+n})\wedge
d(E^{j+n,i}+E^{i+n,j}),&1\leq j\leq n\\\\[8.53581pt]
c_{0}^{i,j}=d(E^{i,i+n})\wedge d(E^{n+j,j}),&1\leq i,j\leq
n\text{~{}~{}except~{}}i=j=n.\\\\[8.53581pt] \end{array}$

######  Lemma.

For $p=2$, the space $H^{2}({\mathfrak{pe}}_{gen}(2n))$ is spanned by the
cocycles (21) (except the cocycles $c_{-1}^{j}$ and symmetric to them
$c_{1}^{j}$ that become trivial), and also the following cocycles:

(22) $\begin{array}[]{ll}c_{-2}^{i}=d(E^{n+i,i})^{\wedge^{2}},&1\leq i\leq
n\\\\[5.69054pt]
\tilde{c}_{-2}^{i,j}=(d(E^{n+i,j}+E^{n+j,i}))^{\wedge^{2}},&1\leq i<j\leq
n\\\\[5.69054pt] \end{array}$

######  Lemma.

For $p=2$, the space $H^{2}({\mathfrak{o}}_{gen}(2n)/{\mathbb{K}}1_{2n})$ is
spanned by the following cocycles

(23) $\begin{array}[]{ll}\text{$\deg=-2$}:&\text{Cocycles as in
\eqref{pogen}}\\\ \text{$\deg=-1$}:&\text{None}\\\
\text{$\deg=0$}:&\text{Cocycles as in \eqref{pogen} and the cocycle
}d(E^{n,2n})\wedge d(E^{2n,n})\\\ \end{array}$

######  Lemma.

For $p=2$, the space
$H^{2}({\mathfrak{p}}{\mathfrak{e}}_{gen}(2n)/{\mathbb{K}}1_{2n})$ is spanned
by the following cocycles

(24) $\begin{array}[]{ll}\text{$\deg=-2$}:&\text{Cocycles as in \eqref{pegen}
and \eqref{pogen}}\\\ \text{$\deg=-1$}:&\text{None}\\\
\text{$\deg=0$}:&\text{Cocycles as in \eqref{pogen} and the cocycle
}\\\\[5.69054pt] \text{for $n\neq 4$}:&d(E^{n,2n})\wedge d(E^{2n,n})\\\
\text{for $n=4$}:&\displaystyle\sum_{i=2,3,4}d(E^{1,i}+E^{i+4,5})\wedge
d(E^{i,1}+E^{5,i+4})+\displaystyle\sum_{i=6,7,8}d(E^{1,i}+E^{i-4,5})\wedge
d(E^{i,1}+E^{5,i-4}).\\\\[5.69054pt] \end{array}$

######  Lemma.

For $p=2$ and ${\mathfrak{g}}={\mathfrak{o}}_{gen}(2n)$ or
${\mathfrak{pe}}_{gen}(2n)$, the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$
is spanned by the following derivative $d=\begin{pmatrix}0&0\\\
0&1_{n}\end{pmatrix}$ for any $n>2$ and also the derivatives

(25) $\begin{array}[]{ll}c_{-1}^{j}=\displaystyle\sum_{1\leq i\leq
n}E^{i,i}\otimes d(E^{n+j,j}),&1\leq j\leq n\\\\[5.69054pt] \end{array}$

######  Lemma.

For $p=2$ and ${\mathfrak{g}}={\mathfrak{o}}_{gen}(2n)/{\mathbb{K}}1_{2n}$ or
${\mathfrak{pe}}_{gen}(2n)/{\mathbb{K}}1_{2n}$, , the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following derivative
$d=\begin{pmatrix}0&0\\\ 0&1_{n}\end{pmatrix}$ for any $n>2$.

######  Lemma.

For $p=2$, the space $H^{2}({\mathfrak{o}}{\mathfrak{p}}(2n))$ is spanned by
the cocycles

(26) $\begin{array}[]{lll}\text{$\deg=-2$}:&\text{None}&\\\
c_{-1}^{j}=&\displaystyle\sum_{i\not=j}d(E^{i,j}+E^{j+n,i+n})\wedge
d(E^{j+n,i}+E^{i+n,j}),&1\leq j\leq n\\\
\text{$\deg=0$}:&\text{None}&\\\\[5.69054pt]
c_{1}^{j}=&\displaystyle\sum_{1\leq i\leq n}d(E^{i,i}+E^{i+n,i+n})\wedge
d(E^{j,n+j}),&1\leq j\leq n\\\\[5.69054pt]
\tilde{c}_{1}^{j}=&\displaystyle\sum_{i\not=j}d(E^{j,i}+E^{i+n,j+n})\wedge
d(E^{i,n+j}+E^{j,n+i})+d(E^{j,j}+E^{n+j,n+j})\wedge d(E^{j,n+j}),&1\leq j\leq
n\\\\[5.69054pt] \text{$\deg=2$}:&\text{Cocycles as in \eqref{pogen}}&\\\
\end{array}$

######  Lemma.

For $p=2$, the space $H^{2}({\mathfrak{pe}}(2n))$ is spanned by the cocycles

(27)
$\begin{array}[]{lll}\text{$\deg=-2$}:&\tilde{c}_{-2}^{i,j}=(d(E^{n+i,j}+E^{n+j,i}))^{\wedge^{2}},&1\leq
i<j\leq n\\\\[5.69054pt] \text{$\deg=-1$}:&\text{None}&\\\
\text{$\deg=0$}:&\text{None}&\\\
\text{$\deg=1$}:&c_{1}^{j}=\displaystyle\sum_{i=1}^{n}d(E^{i,i}+E^{i+n,i+n})\wedge
d(E^{j,n+j}),&1\leq j\leq n\\\\[5.69054pt]
&\tilde{c}_{1}^{j}=\displaystyle\sum_{i\not=j}^{n}d(E^{j,i}+E^{i+n,j+n})\wedge
d(E^{i,n+j}+E^{j,n+i})+d(E^{j,j}+&\\\ &E^{n+j,n+j})\wedge d(E^{j,n+j}),&1\leq
j\leq n\\\\[5.69054pt] \text{$\deg=2$}:&c_{2}=\text{Cocycles as in
\eqref{pogen} and \eqref{pegen}}&\\\ \end{array}$

######  Lemma.

For $p=2$, the space
$H^{1}({\mathfrak{o}}{\mathfrak{p}}(2n);{\mathfrak{o}}{\mathfrak{p}}(2n))$ is
spanned by the following derivatives111Hereafter notation
$\begin{pmatrix}0&0\\\ E^{j,j}&0\end{pmatrix}$ and such like designates a
matrix with only one non-zero element, namely the block represented by a
matrix unit (here $E^{j,j}$).

(28) $\begin{array}[]{ccl}\text{$\deg=-1$}&:&\text{None}\\\
\text{$\deg=0$}&:&\text{as multiplication by the matrix
}d=\begin{pmatrix}0&0\\\ 0&1_{n}\end{pmatrix}\text{ for any $n>2$ and also the
cocycle}\\\\[5.69054pt] &&c_{0}^{2}=\displaystyle\sum_{1\leq i\leq
2n}E^{i,i}\otimes\left(d(E^{1,1}+E^{n+1,n+1})+\cdots+d(E^{n,n}+E^{2n,2n})\right)\\\\[5.69054pt]
\text{$\deg=1$}&:&c_{1}^{j}=\displaystyle\sum_{1\leq i\leq
2n}E^{i,i}d(E^{j,j+n}),\text{~{}~{}for~{}~{}}1\leq j\leq n\text{ and a
cocycle}\\\\[5.69054pt] &&\tilde{c}_{1}^{j}\text{ as multiplication by the
matrix }d_{j}=\begin{pmatrix}0&0\\\ E^{j,j}&0\end{pmatrix}\text{ for any
$n>2$. }\end{array}$

######  Lemma.

For $p=2$, the space $H^{1}({\mathfrak{pe}}(2n);{\mathfrak{pe}}(2n))$ is
spanned by the following derivatives

(29) $\begin{array}[]{ll}\text{$\deg=-1$}:&\text{None}\\\
\text{$\deg=0$}:&\text{as multiplication by the matrix
}d=\begin{pmatrix}0&0\\\ 0&1_{n}\end{pmatrix}\text{ for any $n>2$ and also the
cocycle}\\\\[5.69054pt] &c_{0}^{2}=\displaystyle\sum_{1\leq i\leq
2n}E^{i,i}\otimes\left(d(E^{1,1}+E^{n+1,n+1})+\cdots+d(E^{n,n}+E^{2n,2n})\right)\\\\[5.69054pt]
\text{$\deg=1$}:&\tilde{c}_{1}^{j}\text{ as multiplication by the matrix
}d_{j}=\begin{pmatrix}0&0\\\ E^{j,j}&0\end{pmatrix}\text{ for any $n>2$.
}\end{array}$

######  Lemma.

For $p=2$, the space $H^{2}({\mathfrak{p}}{\mathfrak{o}}{\mathfrak{p}}(2n))$
is spanned by the following cocycles

(30) $\begin{array}[]{ll}\text{$\deg=-2$}:&\text{None}\\\
\text{$\deg=-1$}:&\text{None}\\\ \text{$\deg=0$}:&\text{None for $n\not=4$.
For $n=4$, the cocycle is:}\\\ c_{0}=&d(E^{1,2}+E^{6,5})\wedge
d(E^{2,1}-E^{5,6})+d(E^{2,3}+E^{7,6})\wedge d(E^{3,2}-E^{6,7})\\\\[5.69054pt]
&+d(E^{2,4}+E^{8,6})\wedge d(E^{4,2}-E^{6,8})+d(E^{2,7}+E^{3,6})\wedge
d(E^{7,2}-E^{6,3})\\\\[5.69054pt] &+d(E^{2,8}+E^{4,6})\wedge
d(E^{6,4}-E^{8,2})+d(E^{1,6}+E^{2,5})\wedge d(E^{5,2}-E^{6,1})\\\\[5.69054pt]
c_{1}^{j}=&\displaystyle\sum_{2\leq i\leq n}d(E^{i,i}+E^{i+n,i+n})\wedge
d(E^{j,j+n}),\text{~{}where~{}}1\leq j\leq n.\\\
\text{$\deg=2$}:&\text{Cocycles as in \eqref{pogen}}\end{array}$

######  Lemma.

For $p=2$, the space
$H^{1}({\mathfrak{p}}{\mathfrak{o}}{\mathfrak{p}}(2n);{\mathfrak{p}}{\mathfrak{o}}{\mathfrak{p}}(2n))$
is spanned by the following derivatives

(31) $\begin{array}[]{ll}\text{$\deg=-1$}:&\text{None}\\\
\text{$\deg=0$}:&\text{as multiplication by the matrix
}d=\begin{pmatrix}0&0\\\ 0&1_{n}\end{pmatrix}\text{ for any
$n>2$}\\\\[5.69054pt] \text{$\deg=1$}:&\tilde{c}_{1}^{j}\text{ as
multiplication by the matrix }d_{j}=\begin{pmatrix}0&0\\\
E^{j,j}&0\end{pmatrix}\text{ for any $n>2$ }\end{array}$

######  Lemma.

For $p=2$, the space $H^{2}({\mathfrak{p}}{\mathfrak{p}}{\mathfrak{e}}(2n))$
is spanned by the following cocycles:

(32)
$\begin{array}[]{ll}\text{$\deg=-2$}:&c_{-2}^{i,j}=(d(E^{n+i,j}+E^{n+j,i}))^{\wedge^{2}},\text{~{}~{}where
~{}~{}}1\leq i<j\leq n\\\\[5.69054pt] \text{$\deg=-1$}:&\text{ None }\\\
\text{$\deg=0$}:&\text{None for $n\not=4$. For $n=4$, the cocycle is }\\\
c_{0}=&d(E^{1,2}+E^{6,5})\wedge d(E^{2,1}+E^{5,6})+d(E^{2,3}+E^{7,6})\wedge
d(E^{3,2}+E^{6,7})\\\\[5.69054pt] &+d(E^{2,4}+E^{8,6})\wedge
d(E^{4,2}+E^{6,8})+d(E^{2,7}+E^{3,6})\wedge d(E^{7,2}+E^{6,3})\\\\[5.69054pt]
&+d(E^{2,8}+E^{4,6})\wedge d(E^{6,4}+E^{8,2})+d(E^{1,6}+E^{2,5})\wedge
d(E^{5,2}+E^{6,1})\\\\[5.69054pt] \text{$\deg=1$}:&\displaystyle\sum_{2\leq
i\leq n}d(E^{i,i}+E^{i+n,i+n})\wedge d(E^{j,j+n}),\text{~{}~{}where
~{}~{}}1\leq j\leq n.\\\\[5.69054pt] \text{$\deg=2$}:&\text{Cocycles as in
\eqref{pogen} and \eqref{pegen} }\end{array}$

######  Lemma.

For $p=2$ and ${\mathfrak{g}}={\mathfrak{p}}{\mathfrak{o}}{\mathfrak{p}}(2n)$
or ${\mathfrak{p}}{\mathfrak{p}}{\mathfrak{e}}(2n)$, the space
$H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following derivatives

(33) $\begin{array}[]{ll}\text{$\deg=-1$}:&\text{None}\\\
\text{$\deg=0$}:&\text{as multiplication by the matrix
}d=\begin{pmatrix}0&0\\\ 0&1_{n}\end{pmatrix}\text{ for any
$n>2$}\\\\[5.69054pt] \text{$\deg=1$}:&\tilde{c}_{1}^{j}\text{ as
multiplication by the matrix }d_{j}=\begin{pmatrix}0&0\\\
E^{j,j}&0\end{pmatrix}\text{ for any $n>2$ }\end{array}$

######  Lemma.

For $p=2$, the space $H^{2}({\mathfrak{o}}^{(1)}(2n))$ is spanned by the
following cocycles:

(34) $\begin{array}[]{ll}\text{$\deg=-2$}:&\text{None}\\\\[8.53581pt]
c_{-1}^{j}=&\displaystyle\sum_{i\not=j}d(E^{i,j}+E^{j+n,i+n})\wedge
d(E^{j+n,i}+E^{i+n,j}),\quad 1\leq j\leq n\\\\[8.53581pt]
c_{0}=&\displaystyle\sum_{2\leq i\leq n}d(E^{1,i}+E^{i+n,n+1})\wedge
d(E^{i,1}+E^{n+1,i+n})\\\ &\displaystyle+\sum_{2\leq i<j\leq
n}d(E^{i,n+j}+E^{j,n+i})\wedge d(E^{n+j,i}+E^{n+i,j})\\\\[8.53581pt]
\end{array}$

######  Lemma.

For $p=2$, the space
$H^{1}({\mathfrak{o}}^{(1)}(2n);{\mathfrak{o}}^{(1)}(2n))$ is spanned by the
following derivatives

(35) $\begin{array}[]{ll}\text{$\deg=-1$}:&\tilde{c}_{-1}^{j}\text{ as
multiplication by the matrix }d_{j}=\begin{pmatrix}0&E^{j,j}\\\
0&0\end{pmatrix}\text{ for any $n>2$ }\\\ \text{$\deg=0$}:&\text{as
multiplication by the matrix }d=\begin{pmatrix}0&0\\\
0&1_{n}\end{pmatrix}\text{ for any $n>2$ and also the cocycle}\\\\[5.69054pt]
&c_{0}^{2}=\displaystyle\sum_{1\leq i\leq
2n}E^{i,i}\otimes\left(d(E^{1,1}+E^{n+1,n+1})+\cdots+d(E^{n,n}+E^{2n,2n})\right)\\\\[5.69054pt]
\text{$\deg=1$}:&\tilde{c}_{1}^{j}\text{ as multiplication by the matrix
}d_{j}^{t}=\begin{pmatrix}0&0\\\ E^{j,j}&0\end{pmatrix}\text{ for any $n>2$
}\end{array}$

######  Lemma.

For $p=2$, the space $H^{2}({\mathfrak{p}}{\mathfrak{o}}^{(1)}(2n))$ is
spanned by the following cocycles:

(36) $\begin{array}[]{ll}\text{$\deg=-2$}:&\text{None}\\\\[8.53581pt]
c_{-1}^{j}=&\displaystyle\sum_{i\not=j}d(E^{i,j}+E^{j+n,i+n})\wedge
d(E^{j+n,i}+E^{i+n,j}),\quad 1\leq j\leq n\\\\[8.53581pt]
c_{0}=&\displaystyle\sum_{2\leq i\leq n}d(E^{1,i}+E^{i+n,n+1})\wedge
d(E^{i,1}+E^{n+1,i+n})\displaystyle+\sum_{2\leq i<j\leq
n}d(E^{i,n+j}+E^{j,n+i})\wedge d(E^{n+j,i}+E^{n+i,j})\\\\[8.53581pt]
&\text{For $n=4$, we have one more cocycle }\\\\[8.53581pt]
\tilde{c}_{0}=&d(E^{1,3}+E^{7,5})\wedge
d(E^{3,1}+E^{5,7})+d(E^{1,4}+E^{8,5})\wedge d(E^{4,1}+E^{5,8})\\\
&+d(E^{2,3}+E^{7,6})\wedge d(E^{3,2}+E^{6,7})+d(E^{2,4}+E^{8,6})\wedge
d(E^{4,2}+E^{6,8})\\\\[8.53581pt] &+d(E^{3,8}+E^{4,7})\wedge
d(E^{7,4}+E^{8,3})+d(E^{1,6}+E^{2,5})\wedge d(E^{5,2}+E^{6,1})\\\\[8.53581pt]
\end{array}$

######  Lemma.

For $p=2$, the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$, for
${\mathfrak{g}}={\mathfrak{p}}{\mathfrak{o}}^{(1)}(2n)$, is spanned by the
following derivatives

(37) $\begin{array}[]{ll}\text{$\deg=-1$}:&\tilde{c}_{-1}^{j}\text{ as
multiplication by the matrix }d_{j}=\begin{pmatrix}0&E^{j,j}\\\
0&0\end{pmatrix}\text{ for any $n>2$ }\\\ \text{$\deg=0$}:&\text{as
multiplication by the matrix }d=\begin{pmatrix}0&0\\\
0&1_{n}\end{pmatrix}\text{ for any $n>2$}\\\\[5.69054pt]
\text{$\deg=1$}:&\tilde{c}_{1}^{j}\text{ as multiplication by the matrix
}d_{j}*t=\begin{pmatrix}0&0\\\ E^{j,j}&0\end{pmatrix}\text{ for any $n>2$
}\end{array}$

######  Lemma.

For $p=2$, the space $H^{2}({\mathfrak{s}}{\mathfrak{o}}{\mathfrak{p}}(2n))$
is spanned by the following cocycles:

(38) $\begin{array}[]{ll}\text{$\deg=-2$}:&\text{None for $n\neq 4$. For
$n=4$, we have the cocycle}\\\\[8.53581pt] c_{-2}=&d(E^{7,4}+E^{8,3})\wedge
d(E^{5,2}+E^{6,1})+d(E^{6,3}+E^{7,2})\wedge d(E^{5,4}+E^{8,1})\\\
&+d(E^{6,4}+E^{8,2})\wedge d(E^{5,3}+E^{7,1})\\\\[8.53581pt]
c_{-1}^{j}=&\displaystyle\sum_{i\not=j}d(E^{i,j}+E^{j+n,i+n})\wedge
d(E^{j+n,i}+E^{i+n,j})\quad 1\leq j\leq n\\\ \text{$\deg=0$}:&\text{None
}\\\\[8.53581pt]
c_{1}^{j}=&\displaystyle\sum_{i\not=j}d(E^{j,i}+E^{i+n,j+n})\wedge
d(E^{i,n+j}+E^{j,n+i})\\\\[11.38109pt]
&+\left\\{\begin{array}[]{ll}d(E^{1,1}+E^{2,2}+E^{1+n,1+n}+E^{2+n,2+n})\wedge
d(E^{1,n})&\text{for }i=1\\\
d(E^{n-1,n-1}+E^{n,n}+E^{2n-1,2n-1}+E^{2n,2n})\wedge d(E^{n,2n})&\text{for
}i=n\\\ d(E^{j-1,j-1}+E^{j+1,j+1}+E^{j-1+n,j-1+n}+E^{j+1+n,j+1+n})\wedge
d(E^{j,j+n})&\text{for }j\not=1,n\\\ \end{array}\right.\\\\[8.53581pt]
\text{$\deg=2$}:&\text{ cocycles as in \eqref{pogen}}\end{array}$

######  Lemma.

For $p=2$, the space
$H^{1}({\mathfrak{s}}{\mathfrak{o}}{\mathfrak{p}}(2n);{\mathfrak{s}}{\mathfrak{o}}{\mathfrak{p}}(2n))$
is spanned by the following derivatives

(39) $\begin{array}[]{ll}\text{$\deg=-1$}:&\text{ None}\\\
\text{$\deg=0$}:&\text{as multiplication by the matrix
}d=\begin{pmatrix}0&0\\\ 0&1_{n}\end{pmatrix}\text{ for any
$n>2$}\\\\[5.69054pt] &\text{If $n$ is even, we have the additional cocycle as
multiplication by the matrix }d_{1,1}=\begin{pmatrix}E^{1,1}&0\\\
0&E^{1,1}\end{pmatrix}\\\\[5.69054pt] \text{$\deg=1$}:&c_{1}^{j}\text{ as
multiplication by the matrix }d_{j}=\begin{pmatrix}0&0\\\
E^{j,j}&0\end{pmatrix}\text{ for any $n>2$ }\end{array}$

######  Lemma.

For $p=2$, the space
$H^{2}({\mathfrak{s}}{\mathfrak{o}}{\mathfrak{p}}(4n)/{\mathbb{K}}1_{4n})$ is
spanned by the following cocycles:

(40) $\begin{array}[]{ll}\text{$\deg=-2$}:&\text{None for $n\neq 2$. For
$n=2$, we have the cocycle}\\\\[8.53581pt] c_{-2}=&d(E^{7,4}+E^{8,3})\wedge
d(E^{5,2}+E^{6,1})+d(E^{6,3}+E^{7,2})\wedge d(E^{5,4}+E^{8,1})\\\
&+d(E^{6,4}+E^{8,2})\wedge d(E^{5,3}+E^{7,1})\\\\[8.53581pt]
c_{-1}^{j}=&\displaystyle\sum_{i\not=j}d(E^{i,j}+E^{j+2n,i+2n})\wedge
d(E^{j+2n,i}+E^{i+2n,j})\quad 1\leq j\leq 2n\\\
c_{0}=&\displaystyle\sum_{1\leq i\leq 2n}d(E^{1,i}+E^{2n+i,2n+1})\wedge
d(E^{i,1}+E^{2n+1,2n+i})\\\\[8.53581pt] \text{$\deg=1$}:&\text{None
}\\\\[8.53581pt] \text{$\deg=2$}:&\text{ cocycles as in
\eqref{pogen}}\end{array}$

######  Lemma.

For $p=2$ and
${\mathfrak{g}}={\mathfrak{s}}{\mathfrak{o}}{\mathfrak{p}}(4n)/{\mathbb{K}}1_{4n}$,
the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
derivatives

(41) $\begin{array}[]{ll}\text{$\deg=-2$}:&\text{ None for $n\neq 2$. For
$n=2$, we have the cocycle}\\\ c_{-2}=&(E^{3,8}+E^{4,7})\otimes
d(E^{5,2}+E^{6,1})+(E^{2,7}+E^{3,6})\otimes d(E^{5,4}+E^{8,1})\\\\[8.53581pt]
&+(E^{2,8}+E^{4,6})\otimes d(E^{5,3}+E^{7,1})+(E^{1,6}+E^{2,5})\otimes
d(E^{7,4}+E^{8,3})\\\\[8.53581pt] &+(E^{1,7}+E^{3,5})\otimes
d(E^{6,4}+E^{8,2})+(E^{1,8}+E^{4,5})\otimes d(E^{6,3}+E^{7,2})\\\\[8.53581pt]
\text{$\deg=-1$}:&\text{ None}\\\ \text{$\deg=0$}:&\text{as multiplication by
the matrix }d=\begin{pmatrix}0&0\\\ 0&1_{2n}\end{pmatrix}\text{ for any
$n>1$}\\\\[5.69054pt] &\text{and also the cocycle given as multiplication by
the matrix }d_{1,1}=\begin{pmatrix}E^{1,1}&0\\\
0&E^{1,1}\end{pmatrix}\\\\[5.69054pt] \text{$\deg=1$}:&\text{
None}\end{array}$

######  Lemma.

For $p=2$, the space $H^{2}({\mathfrak{s}}{\mathfrak{p}}{\mathfrak{e}}(2n))$
is spanned by the following cocycles:

(42)
$\begin{array}[]{ll}\text{$\deg=-2$}:&\tilde{c}_{-2}^{i,j}=(d(E^{n+i,j}+E^{n+j,i}))^{\wedge^{2}}\quad
1\leq i<j\leq n\\\\[8.53581pt] &\text{For $n=4$, we have one more cocycle}\\\
&c_{-2}=d(E^{7,4}+E^{8,3})\wedge d(E^{5,2}+E^{6,1})+d(E^{6,3}+E^{7,2})\wedge
d(E^{5,4}+E^{8,1})\\\ &+d(E^{6,4}+E^{8,2})\wedge
d(E^{5,3}+E^{7,1})\\\\[8.53581pt] \text{$\deg=-1$}:&\text{
None}\\\\[8.53581pt] \text{$\deg=0$}:&\text{None }\\\\[8.53581pt]
\text{$\deg=1$}:&c_{1}^{j}=\displaystyle\sum_{i\not=j}d(E^{j,i}+E^{i+n,j+n})\wedge
d(E^{i,n+j}+E^{j,n+i})\\\\[11.38109pt]
&+\left\\{\begin{array}[]{ll}d(E^{1,1}+E^{2,2}+E^{1+n,1+n}+E^{2+n,2+n})\wedge
d(E^{1,n})&\text{for }i=1\\\
d(E^{n-1,n-1}+E^{n,n}+E^{2n-1,2n-1}+E^{2n,2n})\wedge d(E^{n,2n})&\text{for
}i=n\\\ d(E^{j-1,j-1}+E^{j+1,j+1}+E^{j-1+n,j-1+n}+E^{j+1+n,j+1+n})\wedge
d(E^{j,j+n})&\text{for }j\not=1,n\\\ \end{array}\right.\\\\[8.53581pt]
\text{$\deg=2$}:&\text{ cocycles as in \eqref{pogen} and
\eqref{pegen}}\end{array}$

######  Lemma.

For $p=2$ and ${\mathfrak{g}}={\mathfrak{s}}{\mathfrak{p}}{\mathfrak{e}}(2n)$,
the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
derivatives

(43) $\begin{array}[]{ll}\text{$\deg=-1$}:&\text{ None}\\\
\text{$\deg=0$}:&\text{as multiplication by the matrix
}d=\begin{pmatrix}0&0\\\ 0&1_{n}\end{pmatrix}\text{ for any
$n>2$}\\\\[5.69054pt] &\text{If $n$ is even, we have a cocycle as
multiplication by the matrix }d_{1,1}=\begin{pmatrix}E^{1,1}&0\\\
0&E^{1,1}\end{pmatrix}\\\\[5.69054pt] \text{$\deg=1$}:&c_{1}^{j}\text{ as
multiplication by the matrix }d_{j}=\begin{pmatrix}0&0\\\
E^{j,j}&0\end{pmatrix}\text{ for any $n>2$ }\end{array}$

######  Lemma.

For $p=2$, the space
$H^{2}({\mathfrak{s}}{\mathfrak{p}}{\mathfrak{e}}(4n)/{\mathbb{K}}1_{4n})$ is
spanned by the following cocycles:

(44)
$\begin{array}[]{ll}\text{$\deg=-2$}:&\tilde{c}_{-2}^{i,j}=(d(E^{2n+i,j}+E^{2n+j,i}))^{\wedge^{2}}\quad
1\leq i<j\leq 2n\\\\[8.53581pt] &\text{For $n=2$, we have one more cocycle}\\\
c_{-2}=&d(E^{7,4}+E^{8,3})\wedge d(E^{5,2}+E^{6,1})+d(E^{6,3}+E^{7,2})\wedge
d(E^{5,4}+E^{8,1})\\\ &+d(E^{6,4}+E^{8,2})\wedge
d(E^{5,3}+E^{7,1})\\\\[8.53581pt] \text{$\deg=-1$}:&\text{
None}\\\\[8.53581pt] c_{0}=&\displaystyle\sum_{1\leq i\leq
2n}d(E^{1,i}+E^{2n+i,2n+1})\wedge d(E^{i,1}+E^{2n+1,2n+i})\\\\[8.53581pt]
\text{$\deg=1$}:&\text{ None}\\\\[8.53581pt] \text{$\deg=2$}:&\text{ cocycles
as in \eqref{pogen} and \eqref{pegen}}\end{array}$

######  Lemma.

For $p=2$ and
${\mathfrak{g}}={\mathfrak{s}}{\mathfrak{p}}{\mathfrak{e}}(4n)/{\mathbb{K}}1_{4n}$,
the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
derivatives

(45) $\begin{array}[]{ll}\text{$\deg=-2$}:&\text{ None for $n\neq 2$. For
$n=2$, we have the cocycle}\\\ c_{-2}=&(E^{3,8}+E^{4,7})\otimes
d(E^{5,2}+E^{6,1})+(E^{2,7}+E^{3,6})\otimes d(E^{5,4}+E^{8,1})\\\\[8.53581pt]
&+(E^{2,8}+E^{4,6})\otimes d(E^{5,3}+E^{7,1})+(E^{1,6}+E^{2,5})\otimes
d(E^{7,4}+E^{8,3})\\\\[8.53581pt] &+(E^{1,7}+E^{3,5})\otimes
d(E^{6,4}+E^{8,2})+(E^{1,8}+E^{4,5})\otimes d(E^{6,3}+E^{7,2})\\\\[8.53581pt]
\text{$\deg=-1$}:&\text{ None}\\\ \text{$\deg=0$}:&\text{as multiplication by
the matrix }d=\begin{pmatrix}0&0\\\ 0&1_{2n}\end{pmatrix}\text{ for any
$n>1$}\\\\[5.69054pt] &\text{and another cocycle as multiplication by the
matrix }d_{1,1}=\begin{pmatrix}E^{1,1}&0\\\
0&E^{1,1}\end{pmatrix}\\\\[5.69054pt] \text{$\deg=1$}:&\text{
None}\end{array}$

######  Lemma.

For $p=2$, the space $H^{2}({\mathfrak{o}}^{(2)}(2n))$ is spanned by the
following cocycles:

(46) $\begin{array}[]{ll}\text{$\deg=-2$}:&\text{None for $n\neq 4$. For
$n=4$, we have the cocycle}\\\ c_{-2}=&d(E^{7,4}+E^{8,3})\wedge
d(E^{5,2}+E^{6,1})+d(E^{6,3}+E^{7,2})\wedge d(E^{5,4}+E^{8,1})\\\
&+d(E^{6,4}+E^{8,2})\wedge d(E^{5,3}+E^{7,1})\\\\[8.53581pt]
\text{$\deg=-1$}:&c^{j}_{-1}=\displaystyle\sum_{i\not=j}d(E^{i,j}+E^{j+n,i+n})\wedge
d(E^{j+n,i}+E^{i+n,j})\quad 1\leq j\leq n\\\\[8.53581pt] &\text{For $n=4$,
there are also the following $4$ cocycles:}\\\
&c_{-1}^{1}=d(E^{1,2}+E^{6,5})\wedge
d(E^{7,4}+E^{8,3})+d(E^{1,3}+E^{7,5})\wedge d(E^{6,4}+E^{8,2})\\\
&+d(E^{1,4}+E^{8,5})\wedge d(E^{6,3}+E^{7,2})\\\\[8.53581pt]
&c_{-1}^{2}=d(E^{2,3}+E^{7,6})\wedge
d(E^{5,4}+E^{8,1})+d(E^{2,4}+E^{8,6})\wedge d(E^{5,3}+E^{7,1})\\\
&+d(E^{2,1}+E^{5,6})\wedge d(E^{7,4}+E^{8,3})\\\\[8.53581pt]
&c_{-1}^{3}=d(E^{3,4}+E^{8,7})\wedge
d(E^{5,2}+E^{6,1})+d(E^{3,2}+E^{6,7})\wedge d(E^{5,4}+E^{8,1})\\\
&+d(E^{3,1}+E^{5,7})\wedge d(E^{6,4}+E^{8,2})\\\\[8.53581pt]
&c_{-1}^{4}=d(E^{4,3}+E^{7,8})\wedge
d(E^{5,2}+E^{6,1})+d(E^{4,2}+E^{6,8})\wedge d(E^{5,3}+E^{7,1})\\\
&+d(E^{4,1}+E^{5,8})\wedge d(E^{6,3}+E^{7,2})\\\\[8.53581pt]
\text{$\deg=0$}:&c_{0}=\displaystyle\sum_{1\leq i\leq
n}d(E^{1,i}+E^{n+i,n+1})\wedge d(E^{i,1}+E^{n+1,n+i})\\\\[8.53581pt]
&+\displaystyle\sum_{2\leq i<j\leq n}d(E^{i,n+j}+E^{j,n+i})\wedge
d(E^{n+j,i}+E^{n+i,j})\\\\[8.53581pt] &\text{For $n=4$, there are also six
cocycles, given by}\\\ &c_{0}^{1}=d(E^{3,2}-E^{6,7})\wedge
d(E^{4,1}-E^{5,8})+d(E^{4,2}-E^{6,8})\wedge d(E^{3,1}-E^{5,7})\\\
&+d(E^{3,8}-E^{4,7})\wedge d(E^{5,2}+E^{6,1})\\\\[8.53581pt]
&c_{0}^{2}=d(E^{3,4}-E^{8,7})\wedge
d(E^{2,1}-E^{5,6})+d(E^{2,4}-E^{8,6})\wedge d(E^{3,1}-E^{5,7})\\\
&+d(E^{2,7}-E^{3,6})\wedge d(E^{5,4}+E^{8,1})\\\\[8.53581pt]
&c_{0}^{3}=d(E^{4,3}-E^{7,8})\wedge
d(E^{2,1}-E^{5,6})+d(E^{2,3}-E^{7,6})\wedge d(E^{4,1}-E^{5,8})\\\
&+d(E^{2,8}-E^{4,6})\wedge d(E^{5,3}+E^{7,1})\\\\[8.53581pt]
&c_{0}^{4}=d(E^{2,3}-E^{7,6})\wedge
d(E^{1,4}-E^{8,5})+d(E^{2,4}-E^{8,6})\wedge d(E^{1,3}-E^{7,5})\\\
&+d(E^{1,6}-E^{2,5})\wedge d(E^{7,4}+E^{8,3})\\\
&c_{0}^{5}=d(E^{3,4}-E^{8,7})\wedge
d(E^{1,2}-E^{6,5})+d(E^{3,2}-E^{6,7})\wedge d(E^{1,4}-E^{8,5})\\\
&+d(E^{1,7}-E^{3,5})\wedge d(E^{6,4}+E^{8,2})\\\\[8.53581pt]
&c_{0}^{6}=d(E^{4,3}-E^{7,8})\wedge
d(E^{1,2}-E^{6,5})+d(E^{4,2}-E^{6,8})\wedge d(E^{1,3}-E^{7,5})\\\
&+d(E^{1,8}-E^{4,5})\wedge d(E^{6,3}+E^{7,2})\\\\[8.53581pt]
\text{$\deg=1$}:&c^{j}_{1}=\displaystyle\sum_{i\not=j}d(E^{j,i}+E^{i+n,j+n})\wedge
d(E^{i,j+n}+E^{j,i+n})\quad 1\leq j\leq n\\\\[8.53581pt] &\text{For $n=4$,
there also the following cocycles:}\\\ &c_{1}^{1}=d(E^{2,1}+E^{5,6})\wedge
d(E^{4,7}+E^{3,8})+d(E^{3,1}+E^{5,7})\wedge d(E^{4,6}+E^{2,8})\\\
&+d(E^{4,1}+E^{5,8})\wedge d(E^{3,6}+E^{2,7})\\\\[8.53581pt]
&c_{1}^{2}=d(E^{3,2}+E^{6,7})\wedge
d(E^{4,5}+E^{1,8})+d(E^{4,2}+E^{6,8})\wedge d(E^{3,5}+E^{1,7})\\\
&+d(E^{1,2}+E^{6,5})\wedge d(E^{4,7}+E^{3,8})\\\\[8.53581pt]
&c_{1}^{3}=d(E^{4,3}+E^{7,8})\wedge
d(E^{2,5}+E^{1,6})+d(E^{2,3}+E^{7,6})\wedge d(E^{4,5}+E^{1,8})\\\
&+d(E^{1,3}+E^{7,5})\wedge d(E^{4,6}+E^{2,8})\\\\[8.53581pt]
&c_{1}^{4}=d(E^{3,4}+E^{8,7})\wedge
d(E^{2,5}+E^{1,6})+d(E^{2,4}+E^{8,6})\wedge d(E^{3,5}+E^{1,7})\\\
&+d(E^{1,4}+E^{8,5})\wedge d(E^{3,6}+E^{2,7})\\\\[8.53581pt]
\text{$\deg=2$}:&\text{None, except for $n=4$ we have the cocycle}\\\
&c_{2}=d(E^{4,7}+E^{3,8})\wedge d(E^{2,5}+E^{1,6})+d(E^{3,6}+E^{2,7})\wedge
d(E^{4,5}+E^{1,8})\\\ &+d(E^{4,6}+E^{2,8})\wedge
d(E^{3,5}+E^{1,7})\\\\[8.53581pt] \end{array}$

######  Lemma.

For $p=2$, the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$, for
${\mathfrak{g}}={\mathfrak{o}}^{(2)}(2n)$, is spanned by the following
derivatives

(47) $\begin{array}[]{ll}\text{$\deg=-1$}:&\text{as multiplication by the
matrix }d_{i}=\begin{pmatrix}0&E^{i,i}\\\ 0&0\end{pmatrix}\text{ for any $n>2$
}(i=1,\ldots,n)\\\\[5.69054pt] \text{$\deg=0$}:&\text{as multiplication by the
matrix }d=\begin{pmatrix}0&0\\\ 0&1_{n}\end{pmatrix}\text{ for any
$n>2$}\\\\[5.69054pt] &\text{If $n$ is even, there is one more cocycle as
multiplication by the matrix }d_{1,1}=\begin{pmatrix}E^{1,1}&0\\\
0&E^{1,1}\end{pmatrix}\\\\[5.69054pt] \text{$\deg=1$}:&\text{as multiplication
by the matrix }d_{i}^{t}=\begin{pmatrix}0&0\\\ E^{i,i}&0\end{pmatrix}\text{
for any $n>2$ }(i=1,\ldots,n)\\\\[5.69054pt] \end{array}$

######  Lemma.

For $p=2$, the space $H^{2}({\mathfrak{o}}^{(2)}(4n)/{\mathbb{K}}1_{4n})$ is
spanned by the following cocycles:

(48) $\begin{array}[]{ccl}\text{$\deg=-2$}&:&\text{As in the case of
${\mathfrak{o}}^{(2)}(4n)$}\\\ \text{$\deg=-1$}&:&\text{As in the case of
${\mathfrak{o}}^{(2)}(4n)$}\\\ \text{$\deg=0$}&:&\text{As in the case of
${\mathfrak{o}}^{(2)}(4n)$ and one more cocycle}\\\
&&\tilde{c}_{0}=\displaystyle\sum_{1\leq i\leq n}d(E^{1,i}+E^{i+n,n+1})\wedge
d(E^{i,1}+E^{n+1,n+i})\\\ \text{$\deg=1$}&:&\text{As in the case of
${\mathfrak{o}}^{(2)}(4n)$}\\\ \text{$\deg=2$}&:&\text{As in the case of
${\mathfrak{o}}^{(2)}(4n)$}\\\ \end{array}$

######  Lemma.

For $p=2$ and ${\mathfrak{g}}={\mathfrak{o}}^{(2)}(4n)/{\mathbb{K}}1_{4n}$,
the space $H^{1}({\mathfrak{g}};{\mathfrak{g}})$ is spanned by the following
derivatives

(49) $\begin{array}[]{ccl}\text{$\deg=-2$}&:&E^{3,8}\otimes
d(E^{5,2}+E^{6,1})-E^{4,7}\otimes d(E^{5,2}+E^{6,1})+E^{2,7}\otimes
d(E^{5,4}+E^{8,1})\\\ &&-E^{3,6}\otimes d(E^{5,4}+E^{8,1})+E^{2,8}\otimes
d(E^{5,3}+E^{7,1})-E^{4,6}\otimes d(E^{5,3}+E^{7,1})+\\\
&&E^{1,6}\otimes(d(E^{7,4}+E^{8,3})-E^{2,5}\otimes
d(E^{7,4}+E^{8,3})+E^{1,7}\otimes d(E^{6,4}+E^{8,2})\\\ &&-E^{3,5}\otimes
d(E^{6,4}+E^{8,2})+E^{1,8}\otimes d(E^{6,3}+E^{7,2})-E^{4,5}\otimes
d(E^{6,3}+E^{7,2})\\\\[8.53581pt] \text{$\deg=-1$}&:&\text{as multiplication
by the matrix }d_{i}=\begin{pmatrix}0&E^{i,i}\\\ 0&0\end{pmatrix}\text{ for
any $n>2$ }(i=1,\ldots,n)\\\\[5.69054pt] &&\text{For $n=2$ there are also four
cocycles given by}\\\ &&c_{-1}^{1}=E^{2,1}\otimes
d(E^{7,4}+E^{8,3})+E^{5,6}\otimes d(E^{7,4}+E^{8,3})+E^{3,1}\otimes
d(E^{6,4}+E^{8,2})\\\ &&+E^{5,7}\otimes d(E^{6,4}+E^{8,2})+E^{4,1}\otimes
d(E^{6,3}+E^{7,2})+E^{5,8}\otimes d(E^{6,3}+E^{7,2})\\\ &&+E^{3,8}\otimes
d(E^{1,2}-E^{6,5})+E^{4,7}\otimes d(E^{1,2}-E^{6,5})+E^{2,7}\otimes
d(E^{1,4}-E^{8,5})\\\ &&+E^{3,6}\otimes d(E^{1,4}-E^{8,5})+E^{2,8}\otimes
d(E^{1,3}-E^{7,5})+E^{4,6}\otimes d(E^{1,3}-E^{7,5})\\\\[8.53581pt]
&&c_{-1}^{2}=E^{1,4}\otimes d(E^{6,3}+E^{7,2})+E^{8,5}\otimes
d(E^{6,3}+E^{7,2})+E^{2,4}\otimes d(E^{5,3}+E^{7,1})\\\ &&+E^{8,6}\otimes
d(E^{5,3}+E^{7,1})+E^{3,4}\otimes d(E^{5,2}+E^{6,1})+E^{8,7}\otimes
d(E^{5,2}+E^{6,1})\\\ &&+E^{2,7}\otimes d(E^{4,1}-E^{5,8})+E^{3,6}\otimes
d(E^{4,1}-E^{5,8})+E^{1,6}\otimes d(E^{4,3}-E^{7,8})\\\ &&+E^{2,5}\otimes
d(E^{4,3}-E^{7,8})+E^{1,7}\otimes d(E^{4,2}-E^{6,8})+E^{3,5}\otimes
d(E^{4,2}-E^{6,8})\\\\[8.53581pt] &&c_{-1}^{3}=E^{1,2}\otimes
d(E^{7,4}+E^{8,3})+E^{6,5}\otimes d(E^{7,4}+E^{8,3})+E^{3,2}\otimes
d(E^{5,4}+E^{8,1})\\\ &&+E^{6,7}\otimes d(E^{5,4}+E^{8,1})+E^{4,2}\otimes
d(E^{5,3}+E^{7,1})+E^{6,8}\otimes d(E^{5,3}+E^{7,1})\\\ &&+E^{3,8}\otimes
d(E^{2,1}-E^{5,6})+E^{4,7}\otimes d(E^{2,1}-E^{5,6})+E^{1,7}\otimes
d(E^{2,4}-E^{8,6})\\\ &&+E^{3,5}\otimes d(E^{2,4}-E^{8,6})+E^{1,8}\otimes
d(E^{2,3}-E^{7,6})+E^{4,5}\otimes d(E^{2,3}-E^{7,6})\\\\[8.53581pt]
&&c_{-1}^{4}=E^{1,3}\otimes d(E^{6,4}+E^{8,2})+E^{7,5}\otimes
d(E^{6,4}+E^{8,2})+E^{2,3}\otimes d(E^{5,4}+E^{8,1})\\\ &&+E^{7,6}\otimes
d(E^{5,4}+E^{8,1})+E^{4,3}\otimes d(E^{5,2}+E^{6,1})+E^{7,8}\otimes
d(E^{5,2}+E^{6,1})\\\ &&+E^{2,8}\otimes d(E^{3,1}-E^{5,7})+E^{4,6}\otimes
d(E^{3,1}-E^{5,7})+E^{1,6}\otimes d(E^{3,4}-E^{8,7})\\\ &&+E^{2,5}\otimes
d(E^{3,4}-E^{8,7})+E^{1,8}\otimes d(E^{3,2}-E^{6,7})+E^{4,5}\otimes
d(E^{3,2}-E^{6,7})\\\\[8.53581pt] \text{$\deg=0$}&:&\text{as multiplication by
the matrix }d=\begin{pmatrix}0&0\\\ 0&1_{n}\end{pmatrix}\text{ for any
$n>1$}\\\\[5.69054pt] &&\text{and the cocycle as multiplication by the matrix
}d_{1,1}=\begin{pmatrix}E^{1,1}&0\\\ 0&E^{1,1}\end{pmatrix}\\\\[5.69054pt]
&&\text{If $n=2$, there are six cocycles given by}\\\\[5.69054pt]
&&c_{0}^{1}=E^{3,1}\otimes d(E^{2,4}-E^{8,6})+E^{5,7}\otimes
d(E^{2,4}-E^{8,6})+E^{4,1}\otimes d(E^{2,3}-E^{7,6})\\\ &&+E^{5,8}\otimes
d(E^{2,3}-E^{7,6})+E^{3,2}\otimes d(E^{1,4}-E^{8,5})+E^{6,7}\otimes
d(E^{1,4}-E^{8,5})\\\ &&+E^{4,2}\otimes d(E^{1,3}-E^{7,5})+E^{6,8}\otimes
d(E^{1,3}-E^{7,5})+E^{3,8}\otimes d(E^{1,6}-E^{2,5})\\\ &&+E^{4,7}\otimes
d(E^{1,6}-E^{2,5})++E^{5,2}\otimes d(E^{7,4}+E^{8,3})+E^{6,1}\otimes
d(E^{7,4}+E^{8,3})\\\\[8.53581pt] &&c_{0}^{2}=(E^{2,4}\otimes
d(E^{1,3}-E^{7,5})+E^{8,6}\otimes d(E^{1,3}-E^{7,5})+E^{3,4}\otimes
d(E^{1,2}-E^{6,5})\\\ &&+E^{8,7}\otimes d(E^{1,2}-E^{6,5})+E^{2,1}\otimes
d(E^{4,3}-E^{7,8})+E^{5,6}\otimes d(E^{4,3}-E^{7,8})\\\ &&+E^{3,1}\otimes
d(E^{4,2}-E^{6,8})+E^{5,7}\otimes d(E^{4,2}-E^{6,8})+E^{2,7}\otimes
d(E^{1,8}-E^{4,5})\\\ &&+E^{3,6}\otimes d(E^{1,8}-E^{4,5})+E^{5,4}\otimes
d(E^{6,3}+E^{7,2})+E^{8,1}\otimes d(E^{6,3}+E^{7,2})\\\\[8.53581pt]
&&c_{0}^{3}=E^{2,3}\otimes d(E^{1,4}-E^{8,5})+E^{7,6}\otimes
d(E^{1,4}-E^{8,5})+E^{2,1}\otimes d(E^{3,4}-E^{8,7})\\\ &&+E^{5,6}\otimes
d(E^{3,4}-E^{8,7})+E^{4,1}\otimes d(E^{3,2}-E^{6,7})+E^{5,8}\otimes
d(E^{3,2}-E^{6,7})\\\ &&+E^{4,3}\otimes d(E^{1,2}-E^{6,5})+E^{7,8}\otimes
d(E^{1,2}-E^{6,5})+E^{2,8}\otimes d(E^{1,7}-E^{3,5})\\\ &&+E^{4,6}\otimes
d(E^{1,7}-E^{3,5})+E^{5,3}\otimes d(E^{6,4}+E^{8,2}+E^{7,1}\otimes
d(E^{6,4}+E^{8,2})\\\\[8.53581pt] \end{array}$ (50)
$\begin{array}[]{lcl}&&c_{0}^{4}=(E^{1,3}\otimes
d(E^{4,2}-E^{6,8})+E^{7,5}\otimes d(E^{4,2}-E^{6,8})+E^{1,4}\otimes
d(E^{3,2}-E^{6,7})\\\ &&+E^{8,5}\otimes d(E^{3,2}-E^{6,7})+E^{2,3}\otimes
d(E^{4,1}-E^{5,8})+E^{7,6}\otimes d(E^{4,1}-E^{5,8})\\\ &&+E^{2,4}\otimes
d(E^{3,1}-E^{5,7})+E^{8,6}\otimes d(E^{3,1}-E^{5,7})+E^{1,6}\otimes
d(E^{3,8}-E^{4,7})\\\ &&+E^{2,5}\otimes d(E^{3,8}-E^{4,7})+E^{7,4}\otimes
d(E^{5,2}+E^{6,1})+E^{8,3}\otimes d(E^{5,2}+E^{6,1})\\\\[8.53581pt]
&&c_{0}^{5}=E^{1,2}\otimes d(E^{4,3}-E^{7,8})+E^{6,5}\otimes
d(E^{4,3}-E^{7,8})+E^{1,4}\otimes d(E^{2,3}-E^{7,6})\\\ &&+E^{8,5}\otimes
d(E^{2,3}-E^{7,6})+E^{3,4}\otimes d(E^{2,1}-E^{5,6})+E^{8,7}\otimes
d(E^{2,1}-E^{5,6})\\\ &&+E^{3,2}\otimes d(E^{4,1}-E^{5,8})+E^{6,7}\otimes
d(E^{4,1}-E^{5,8})+E^{1,7}\otimes d(E^{2,8}-E^{4,6})\\\ &&+E^{3,5}\otimes
d(E^{2,8}-E^{4,6})+E^{6,4}\otimes d(E^{5,3}+E^{7,1})+E^{8,2}\otimes
d(E^{5,3}+E^{7,1})\\\\[8.53581pt] &&c_{0}^{6}=E^{1,2}\otimes
d(E^{3,4}-E^{8,7})+E^{6,5}\otimes d(E^{3,4}-E^{8,7})+E^{1,3}\otimes
d(E^{2,4}-E^{8,6})\\\ &&+E^{7,5}\otimes d(E^{2,4}-E^{8,6})+E^{4,2}\otimes
d(E^{3,1}-E^{5,7})+E^{6,8}\otimes d(E^{3,1}-E^{5,7})\\\ &&+E^{4,3}\otimes
d(E^{2,1}-E^{5,6})+E^{7,8}\otimes d(E^{2,1}-E^{5,6})+E^{1,8}\otimes
d(E^{2,7}-E^{3,6})\\\ &&+E^{4,5}\otimes d(E^{2,7}-E^{3,6})+E^{6,3}\otimes
d(E^{5,4}+E^{8,1})+E^{7,2}\otimes d(E^{5,4}+E^{8,1})\\\\[8.53581pt]
\text{$\deg=1$}&:&\text{as multiplication by the matrix
}d_{i}=\begin{pmatrix}0&0\\\ E^{i,i}&0\end{pmatrix}\text{ for any $n>2$
}(i=1,\ldots,n)\\\\[5.69054pt] &&\text{For $n=2$, there are also four cocycles
given by}\\\ &&c_{1}^{1}=E^{1,2}\otimes d(E^{3,8}-E^{4,7})+E^{6,5}\otimes
d(E^{3,8}-E^{4,7})+E^{1,3}\otimes d(E^{2,8}-E^{4,6})\\\ &&+E^{7,5}\otimes
d(E^{2,8}-E^{4,6})+E^{1,4}\otimes d(E^{2,7}-E^{3,6})+E^{8,5}\otimes
d(E^{2,7}-E^{3,6})\\\ &&+E^{7,4}\otimes d(E^{2,1}-E^{5,6})+E^{8,3}\otimes
d(E^{2,1}-E^{5,6})+E^{6,3}\otimes d(E^{4,1}-E^{5,8})\\\ &&+E^{7,2}\otimes
d(E^{4,1}-E^{5,8})+E^{6,4}\otimes d(E^{3,1}-E^{5,7})+E^{8,2}\otimes
d(E^{3,1}-E^{5,7})\\\\[8.53581pt] &&c_{1}^{2}=E^{2,3}\otimes
d(E^{1,8}-E^{4,5})+E^{7,6}\otimes d(E^{1,8}-E^{4,5})+E^{2,4}\otimes
d(E^{1,7}-E^{3,5})\\\ &&+E^{8,6}\otimes d(E^{1,7}-E^{3,5})+E^{2,1}\otimes
d(E^{3,8}-E^{4,7})+E^{5,6}\otimes d(E^{3,8}-E^{4,7})\\\ &&+E^{7,4}\otimes
d(E^{1,2}-E^{6,5})+E^{8,3}\otimes d(E^{1,2}-E^{6,5})+E^{5,3}\otimes
d(E^{4,2}-E^{6,8})\\\ &&+E^{7,1}\otimes d(E^{4,2}-E^{6,8})+E^{5,4}\otimes
d(E^{3,2}-E^{6,7})+E^{8,1}\otimes d(E^{3,2}-E^{6,7})\\\\[8.53581pt]
&&c_{1}^{3}=E^{3,4}\otimes d(E^{1,6}-E^{2,5})+E^{8,7}\otimes
d(E^{1,6}-E^{2,5})+E^{3,1}\otimes d(E^{2,8}-E^{4,6})\\\ &&+E^{5,7}\otimes
d(E^{2,8}-E^{4,6})+E^{3,2}\otimes d(E^{1,8}-E^{4,5})+E^{6,7}\otimes
d(E^{1,8}-E^{4,5})\\\ &&+E^{6,4}\otimes d(E^{1,3}-E^{7,5})+E^{8,2}\otimes
d(E^{1,3}-E^{7,5})+E^{5,2}\otimes d(E^{4,3}-E^{7,8})\\\ &&+E^{6,1}\otimes
d(E^{4,3}-E^{7,8})+E^{5,4}\otimes d(E^{2,3}-E^{7,6})+E^{8,1}\otimes
d(E^{2,3}-E^{7,6})\\\\[8.53581pt] &&c_{1}^{4}=E^{4,1}\otimes
d(E^{2,7}-E^{3,6})+E^{5,8}\otimes d(E^{2,7}-E^{3,6})+E^{4,2}\otimes
d(E^{1,7}-E^{3,5})\\\ &&+E^{6,8}\otimes d(E^{1,7}-E^{3,5})+E^{4,3}\otimes
d(E^{1,6}-E^{2,5})+E^{7,8}\otimes d(E^{1,6}-E^{2,5})\\\ &&+E^{6,3}\otimes
d(E^{1,4}-E^{8,5})+E^{7,2}\otimes d(E^{1,4}-E^{8,5})+E^{5,2}\otimes
d(E^{3,4}-E^{8,7})\\\ &&+E^{6,1}\otimes d(E^{3,4}-E^{8,7})+E^{5,3}\otimes
d(E^{2,4}-E^{8,6})+E^{7,1}\otimes d(E^{2,4}-E^{8,6})\\\\[8.53581pt]
\text{$\deg=2$}&:&E^{7,4}\otimes
d(E^{1,6}-E^{2,5})+E^{8,3}\otimes(d(E^{1,6}-E^{2,5})+E^{6,3}\otimes
d(E^{1,8}-E^{4,5})\\\ &&+E^{7,2}\otimes d(E^{1,8}-E^{4,5})+E^{6,4}\otimes
d(E^{1,7}-E^{3,5})+E^{8,2}\otimes d(E^{1,7}-E^{3,5})\\\ &&+E^{5,2}\otimes
d(E^{3,8}-E^{4,7})+E^{6,1}\otimes d(E^{3,8}-E^{4,7})+E^{5,3}\otimes
d(E^{2,8}-E^{4,6})\\\ &&+E^{7,1}\otimes d(E^{2,8}-E^{4,6})+E^{5,4}\otimes
d(E^{2,7}-E^{3,6})+E^{8,1}\otimes d(E^{2,7}-E^{3,6})\end{array}$

## References

  * [BGP] Benkart G., Gregory Th., Premet A., _The recognition theorem for graded Lie algebras in prime characteristic_. American Mathematical Society, 2009, 145 pp.
  * [BGLL] Bouarroudj S., Grozman P., Lebedev A., Leites D., Divided power (co)homology. Presentations of simple finite dimensional modular Lie superalgebras with Cartan matrix. Homology, Homotopy and Applications, Vol. 12 (2010), No. 1, 237–278; arXiv:0911.0243
  * [BS] Bouarroudj S., Eick B., Grozman P., Lebedev A., Leites D., Shchepochkina I., Simple prolongs of non-positive parts of graded Lie algebras without Cartan matrix in characteristic $2$; arXiv:??
  * [BGLLSq] Bouarroudj S., Grozman P., Lebedev A., Leites D., Shchepochkina I., Restricted simple Lie algebras and their queerifications in characteristic $2$; arXiv:??
  * [BLLS] Bouarroudj S., Lebedev A., Leites D., Shchepochkina I., Deforms of Lie algebras in characteristic 2: Semi-trivial for Jurman algebras, non-trivial for Kaplansky algebras; arXiv:1301.2781
  * [BLW] Bouarroudj S., Lebedev A., Wagemann F., Deformations of the Lie algebra ${\mathfrak{o}}(5)$ in characteristics $3$ and $2$. Mathem. Notes, 2011, 89:6, 777–791; arXiv:0909.3572
  * [BGL1] Bouarroudj S., Grozman P., Leites D., Classification of simple finite dimensional modular Lie superalgebras with Cartan matrix. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 5 (2009), 060, 63 pages; arXiv:math.RT/0710.5149
  * [BGL2] Bouarroudj S., Grozman P., Leites D., Infinitesimal deformations of symmetric simple modular Lie algebras and Lie superalgebras. arXiv:0807.3054
  * [BGL3] Bouarroudj S., Grozman P., Leites D., New simple modular Lie superalgebras as generalized prolongs. Func. Anal. Appl., Vol. 42, No. 3, pp. 1–9 (Functional Analysis and Its Applications, Vol. 42, No. 3, 2008, 161–168); arXiv:math.RT/0704.0130
  * [BL] Bouarroudj S., Leites D., Simple Lie superalgebras and non-integrable distributions in characteristic $p$. Zapiski nauchnyh seminarov POMI, t. 331 (2006), 15–29; Reprinted in J. Math. Sci. (NY) 2007; arXiv:math.RT/0606682
  * [Br3] Brown, G., Properties of a 29-dimensional simple Lie algebra of characteristic three, Math. Ann. 261 (1982), 487–492
  * [CCLL] Chapovalov D., Chapovalov M., Lebedev A., Leites D. The classification of almost affine (hyperbolic) Lie superalgebras // 2010, vol. 17, Special issue in memory of F. Berezin, 103–161; arXiv:0906.1860
  * [Ch] Chebochko, N. G. Deformations of classical Lie algebras with a homogeneous root system in characteristic two. I. (Russian) Mat. Sb. 196 (2005), no. 9, 125–156; translation in Sb. Math. 196 (2005), no. 9-10, 1371–1402
  * [ChG2] Chebochko, N. G. Deformations of Lie algebra of type $G_{2}$ in characteristic $2$. The IV International Algebraic Conference, dedicated to Yu. I. Merzlyakov (1940-1995) 60-th birthday anniversary. August 7 – August 11, 2000, Novosibirsk, Russia (in Russian)
  * [Dj] Djoković, D. Ž.; Hochschild, G. Semisimplicity of $2$-graded Lie algebras. II. Illinois J. Math. 20 (1976), no. 1, 134–143. Djoković, D. Ž. Classification of some $2$-graded Lie algebras. J. Pure Appl. Algebra 7 (1976), no. 2, 217–230. Djoković, D. Ž. Isomorphism of some simple $2$-graded Lie algebras. Canad. J. Math. 29 (1977), no. 2, 289–294.
  * [DK] Dzhumadil’daev, A.S.; Kostrikin, A.I. Deformations of the Lie Algebra $W_{1}(m)$ (Russian). Trudy Mat. Inst. Steklov 1978, 148, 141–155.
  * [Dz] Dzhumadil’daev, A.S. On deformations of classical simple Lie algebras (Russian) Uspehi Mat. Nauk 31 (1976), 211–212
  * [Dz1] Dzhumadil’daev, A.S. Deformations of the Lie Algebras $W_{n}(m)$ (Russian). Mat. Sbornik 1989, 180, 168–186
  * [Dz2] Dzhumadil’daev, A.S. Cohomology of truncated co-induced representations of Lie algebras of positive characteristic. (Russian) Mat. Sb. 180 (1989), no. 4, 456–468, 559; translation in Math. USSR-Sb. 66 (1990), no. 2, 461–473
  * [Dz3] Dzhumadil’daev, A.S. On the cohomology of modular Lie algebras. Math. USSR-Sb 47 (1984), N1, 127-143 Mat. Sb. 119 (1982), no. 1, 132–149
  * [Dz4] Dzhumadil’daev, A.S. Central extensions and invariant forms of Cartan Type Lie algebras of positive characteristic, Funct.Anal.Appl. 18(1984), No.4, 331–332.
  * [Dz5] Dzhumadil’daev, A.S. Central extensions of Zassenhaus algebra and their irreducible representations, Math.USSR Sb., 54(1986), p.457–474.
  * [Dz6] Dzhumadil’daev, A.S. Cohomology and nonsplit extensions of modular Lie algebras, Contemp.Math. 131(1992), part 2, p.31–43.
  * [Dz7] Dzhumadil’daev, A.S. Central extensions of infinite-dimensional Lie algebras, Funct.Anal.Appl., 26(1992), No.4, 246–253.
  * [Dz8] Dzhumadil’daev, A.S. Odd central extensions of Lie superalgebras, Funct.Anal. Appl.29(1995), No.3, 69–72
  * [DzU] Dzhumadil’daev, A.S., Umirbaev U., Nonsplit extensions of Cartan type Lie algebra $W_{2}(m)$ , Math. Sbornik. 186(1995), No.4, 527–554.
  * [DzZ] Dzhumadil’daev, A.S., Zusmanovich P., Commutative 2-cocycles on Lie algebras. J. Algebra 324 (2010), 732–748; arXiv:0907.4780v4
  * [Ei] Eick, B., Some new simple Lie algebras in characteristic 2. J. Symb. Comput. 45 (2010), N9, 943–951
  * [FS] Farnsteiner, R.; Strade, H. Shapiro’s lemma and its consequences in the cohomology theory of modular Lie algebras. Math. Z. 206 (1991), no. 1, 153–168
  * [FF] Fialowski, A.; Fuchs, D. Singular deformations of Lie algebras. Example: deformations of the Lie algebra $L_{1}$. Topics in singularity theory, 77–92, Amer. Math. Soc. Transl. Ser. 2, 180, Amer. Math. Soc., Providence, RI, 1997
  * [FG] Frohardt D., Griess R.L. Jr., Automorphisms of modular Lie algebras, Nova J. Algebra and Geometry, v.1, 1992, no.4, 339–345
  * [FL] Fuchs D., Lang L., Massey products and deformations; arXiv:q-alg/9602024
  * [Fu] Fuks (Fuchs) D., Cohomology of infinite dimensional Lie algebras, Consultants Bureau, NY, 1986
  * [FH] Fulton, W., Harris, J., Representation theory. A first course. Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991. xvi+551 pp.
  * [Gr] Grozman P., SuperLie, http://www.equaonline.com/math/SuperLie
  * [GL1] Grozman P., Leites D., Defining relations for classical Lie superalgebras with Cartan matrix, Czech. J. Phys., Vol. 51, 2001, no. 1, 1–22; arXiv:hep-th/9702073
  * [GL4] Grozman P., Leites D., Structures of $G(2)$ type and nonintegrable distributions in characteristic $p$. Lett. Math. Phys. 74 (2005), no. 3, 229–262; arXiv:math.RT/0509400
  * [HS] Hoyt, C.; Serganova, V. Classification of finite-growth general Kac-Moody superalgebras. Comm. Algebra 35 (2007), no. 3, 851–874
  * [ILL] Iyer U., Lebedev A., Leites D., Prolongs of orthogonal Lie (super)algebras in characteristic $2$. J. Nonlinear Math. Phys., v.17 (2010), Special issue in memory of F. Berezin, 253–309;
  * [K] Kac V. Infinite-dimensional Lie algebras. Third edition. Cambridge University Press, Cambridge, 1990. xxii+400 pp.
  * [KWK] Kac, V. G. Corrections to: “Exponentials in Lie algebras of characteristic $p$” [Izv. Akad. Nauk SSSR 35 (1971), no. 4, 762–788; MR0306282 (46 #5408)] by B. Yu. Veisfeiler and Kac. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 4, 224; translation in Russian Acad. Sci. Izv. Math. 45 (1995), no. 1, 229
  * [Kapp] Kaplansky I., Graded Lie algebras, preprints, Univ. Chicago, Chicago, 1975 http://justpasha.org/math/links/subj/lie/kaplansky
  * [KKCh] Kirillov, S. A.; Kuznetsov, M. I.; Chebochko, N. G. Deformations of a Lie algebra of type $G_{2}$ of characteristic three. Russian Math. (Iz. VUZ) 44 (2000), no. 3, 31–36
  * [Kos] Kostrikin, A. I. A parametric family of simple Lie algebras. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 34 1970 744–756
  * [KK] Kostrikin, A. I.; Kuznetsov, M. I. On deformations of classical Lie algebras of characteristic three. (Russian) Dokl. Akad. Nauk 343 (1995), no. 3, 299–301.
  * [KCh] Kuznetsov, M. I.; Chebochko, N. G. Deformations of classical Lie algebras. (Russian) Mat. Sb. 191 (2000), no. 8, 69–88; translation in Sb. Math. 191 (2000), no. 7-8, 1171–1190
  * [LeD] Lebedev A., Simple modular Lie superalgebras. Ph.D. thesis. Leipzig University, July, 2008.
  * [LeP] Lebedev A., Analogs of the orthogonal, Hamiltonian, Poisson, and contact Lie superalgebras in characteristic $2$. J. Nonlinear Math. Phys., vol. 17, Special issue in memory of F. Berezin, 2010, 217–251
  * [LL] Lebedev A., Leites D., (with an appendix by Deligne P.) On realizations of the Steenrod algebras, J. of Prime Research in Mathematics, v. 2 no. 1 (2006), 1–13
  * [L0] Leites D. (ed.) Seminar on Supersymmetries, MCCME, Moscow, 2008 (in Russian)
  * [LCh] Leites D. (ed.) Chevalley supergroups and simple modular Lie superalgebras(S. Bouarroudj, B. Clarke, P. Grozman, A. Lebedev, D. Leites, I. Shchepochkina), ca 350 pp. WORK IN PROGRESS
  * [Ltow] Leites D., Towards classification of simple finite dimensional modular Lie superalgebras in characteristic $p$. J. Prime Res. Math., v. 3, 2007, 101–110; arXiv:0710.5638
  * [LSh] Leites D., Shchepochkina I., Classification of the simple Lie superalgebras of vector fields, preprint MPIM-2003-28 (http://www.mpim-bonn.mpg.de)
  * [LLg] Liu, Dong; Lin, Lei. On the variations of $G_{2}$. Chin. Ann. Math. 24B (2003), N3, 387–294
  * [NW] Neeb K.-H., Wagemann F., The second cohomology of current algebras of general Lie algebras, Canad. J. Math. 60 (2008), 892–922; arXiv:math/0511260.
  * [Ru] Rudakov, A. N. Deformations of simple Lie algebras. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1113–1119
  * [Shen1] Shen, Guang Yu, Variations of the classical Lie algebra $G_{2}$ in low characteristics. Nova J. Algebra Geom. 2 (1993), no. 3, 217–243.
  * [Ser] Serganova V., Automorphisms of simple Lie superalgebras. Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 3, 585–598; (Russian) English translation: Math. USSR-Izv. 24 (1985), no. 3, 539–551
  * [S] Strade, H. Simple Lie algebras over fields of positive characteristic. I. Structure theory. de Gruyter Expositions in Mathematics, 38. Walter de Gruyter & Co., Berlin, 2004\. viii+540 pp.
  * [St] Steinberg, R. Lectures on Chevalley groups. Notes prepared by John Faulkner and Robert Wilson. Yale University, New Haven, Conn., 1968\. iii+277 pp.
  * [Vi] Viviani F., Deformations of Simple Restricted Lie Algebras I, II. Journal of Algebra, 320:12 (2008), 4102–??; 213 (2009), 1702–1721; arXiv:math.RA/0612861, math.RA/0702499; Deformations of the restricted Melikian Lie algebra. Comm. Algebra 37 (2009), no. 7, 1850–1872; arXiv:math.RA/0702594; Restricted simple Lie algebras and their infinitesimal deformations; arXiv:math.RA/0702755
  * [Vi1] Viviani F., Simple finite group schemes and their infinitesimal deformations. Rend. Sem. Mat. Univ. Politec. Torino Vol. 68 (2010), 171–182
  * [WK] Weisfeiler B., Kac V. Exponentials in Lie algebras of characteristic $p$. Izv. Akad. Nauk SSSR 35 (1971), no. 4, 762–788