If $L$ is a Lie algebra, a subspace $B$ of $L$ is called an \emph{inner ideal} if $[B,[B,L]]\subset B$. This notion is inspired in Jordan algebras and it dues to [1], which used it to reconstruct the geometry defined by Tits from the corresponding Chevalley group. Soon, [2] began a sistematic study of inner ideals of Lie algebras with a view in an Artinian theory for Lie algebras (no restrictions on the dimension or on the characteristic of the field). A good compilation from the algebraic approach can be found in the recent monograph [3].
In this poster, we clasify abelian inner ideals of the finite-dimensional simple real Lie algebras. Note that the classification of the abelian inner ideals of the finite-dimensional simple complex Lie algebras was previously obtained in [4], which provided a concrete description up to automorphisms of these inner ideals in terms of roots. Both classifications are related, since clearly if $B$ is an inner ideal of a real algebra $L$, then the complexification $B^\mathbb C=B\otimes_{\mathbb R}\mathbb C$ is an inner ideal of $L^\mathbb C
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