Triple automorphisms of simple Lie algebras.

An invertible linear map φ on a Lie algebra L is called a triple automorphism of it if φ([ x, [ y, z]]) = [ φ( x), [ φ( y), φ( z)]] for ∀ x, y, z ∈ L. Let g be a finite-dimensional simple Lie algebra of rank l defined over an algebraically closed field F of characteristic zero, p an arbitrary parabolic subalgebra of g. It is shown in this paper that an invertible linear map φ on p is a triple automorphism if and only if either φ itself is an automorphism of p or it is the composition of an automorphism of p and an extremal map of order 2. [ABSTRACT FROM AUTHOR]