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simple Lie algebras; parabolic subalgebras; triple automorphisms of Lie algebras
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 $2$.
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