## Some Results on the Classification of Triple Systems |

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a e k algebra system Aut L Aut Aut(S,a Aut*L Aut+L automorphism groups bijective correspondence central subgroup Chapter characteristic zero classes of k-forms classes of polarized cohomology correspondence between elements dim ker direct sum dual Dynkin diagram elements of H Faulkner diagram follows from Lemma g e G G-group Galois descent Galois group h e H H G,A Hence homomorphism inner derivations irreducible components irreducible L-modules irreducible submodules Jordan pair Jordan triple system k-isomorphism k-scalar isomorphism classes Lemma 4.l Lie algebra Lie module triple Lie triple systems module triple system nonassociative algebra nondegenerate nonzero normal subgroup polarization preserving isomorphism polarized k-forms preserving isomorphism classes Proof scalar Schur's lemma semisimple short exact sequences simple Lie triple standard embedding subalgebra Suppose t e Aut t e ker Theorem 5.l0 triple product twisted Lie module x,y e y,z e yes Aut