Triple automorphisms of simple Lie algebras
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 1007-1016.

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An invertible linear map $\varphi $ on a Lie algebra $L$ is called a triple automorphism of it if $\varphi ([x,[y,z]])=[\varphi (x),[ \varphi (y),\varphi (z)]]$ for $\forall x, y, z\in L$. Let $\frak {g}$ be a finite-dimensional simple Lie algebra of rank $l$ defined over an algebraically closed field $F$ of characteristic zero, $\mathfrak {p}$ an arbitrary parabolic subalgebra of $\mathfrak {g}$. It is shown in this paper that an invertible linear map $\varphi $ on $\mathfrak {p}$ is a triple automorphism if and only if either $\varphi $ itself is an automorphism of $\mathfrak {p}$ or it is the composition of an automorphism of $\mathfrak {p}$ and an extremal map of order $2$.
DOI : 10.1007/s10587-011-0043-9
Classification : 17B20, 17B30, 17B40
Mots-clés : simple Lie algebras; parabolic subalgebras; triple automorphisms of Lie algebras 
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Wang, Dengyin; Yu, Xiaoxiang. Triple automorphisms of simple Lie algebras. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 1007-1016. doi : 10.1007/s10587-011-0043-9. https://geodesic-test.mathdoc.fr/articles/10.1007/s10587-011-0043-9/

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