On the structure of split involutive Lie algebras

Abstract

We study the structure of arbitrary split involutive Lie algebras. We show that any of such algebras $L$ is of the form $L={\mathcal U} +\sum_{j}I_{j}$ with ${\mathcal U}$ a subspace of the involutive abelian Lie subalgebra $H$ and any $I_{j}$ a well described involutive ideal of $L$ satisfying $[I_j,I_k]=0$ if $j\neq k$. Under certain conditions, the simplicity of $L$ is characterized and it is shown that $L$ is the direct sum of the family of its minimal involutive ideals, each one being a simple split involutive Lie algebra.