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Let $L$ be classical Lie algebra of type $A_5$ over field of characteristic 2; let $M$ be the quotient $L/Z(L)$ modulo its center $Z(L)$.

What about the group of automorphisms of M?

Does anybody know the answer or at least some related papers where automorphisms of classical Lie algebras over fields of prime characteristic are being studied?

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The answer is fairly classical (and no longer really at research level), but the most natural setting for it is the more general study of Lie algebras obtained by reduction mod $p$ from Chevalley's integral form of a simple Lie algebra over $\mathbb{C}$. The basic source to look at is a 1961 paper by Steinberg, now freely accessible online here. Keep in mind that in the type $A_n$ situation, you are starting with a special linear Lie algebra $\mathfrak{sl}_{n+1}$ having an obvious Chevalley basis over $\mathbb{Z}$. Precisely when $n$ is odd (such as your case $n=5$), the resulting algebra over any field of characteristic 2 has as center the 1-dimensional space of scalar matrices. Modulo the center, the resulting Lie algebra is simple over virtually any field with only small exceptions in low rank when the field is small.

Steinberg's approach has the advantage of applying uniformly to all Lie types. In type $A_n$ the automorphism group consists of inner automorphisms (those coming by conjugation from the corresponding projective special linear group) composed with powers of the unique graph automorphism of order 2 if $n>1$.

Note that all of this is discussed in Chapter III of George Seligman's 1967 Ergebnisse survey Modular Lie Algebras (Springer-Verlag), with further background references to classical work in characteristic 0.

[Question: Is there a reason for your interest in the special case $n=5$?]

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