On Twisted Inner Derivation Triple Systems And Symmetric Invariant Bilinear Forms On Them

As twisted inner derivation triple systems (TJDTS) take Lie triple systems (L.t.s.) as well as some other kinds of triple systems as their special cases, an ensuing question is whether the standard results valid in L.t.s. still hold for TIDTS. Proved in the first half of this work is the assertion that the standard imbedding S(M) of a simple TIDTS (M,{ , , )) is either simple or a direct sum of two simple ideals, with (M,{ , , }) being a L.t.s. of a Lie algebra when the latter case occurs. The second half of the work is devoted to the possible extention of a bilinear form on a TIDTS. Given a naturally-derived concept of invariance, a symmetric invariant bilinear form cI~ ( , ) on a TIDTS (M,{ , , }) can be uniquely extended to a same-natured form cI~ ( , ) on the standard imbedding S(M). cI~ (,) is non degenerate iff c1(,)is.