Lie algebras with S4-action and structurable algebras

The normal symmetric triality algebras (STAʹs) and the normal Lie related triple algebras (LRTAʹs) have been recently introduced by the second author, in connection with the principle of triality. It turns out that the unital normal LRTAʹs are precisely the structurable algebras extensively studied by Allison. It will be shown that the normal STAʹs (respectively LRTAʹs) are the algebras that coordinatize those Lie algebras whose automorphism group contains a copy of the alternating (respectively symmetric) group of degree 4.