Non-degenerate invariant bilinear forms on nonassociative triple systems (Q2483729)

A bilinear form \(f\) on a triple system \(\mathcal T\) over a field \(\mathcal F\) is said to be invariant if and only if \(f(\langle abc\rangle; d) = f(a; \langle dcb\rangle = f(c; \langle bad\rangle)\) for all \(a, b, c, d \in \mathcal T\). The authors investigate the uniqueness of decompositions of a finite-dimensional triple system \(\mathcal T\) as a direct sum of indecomposable ideals. For a triple system \(\mathcal T\) admitting a nondegenerate invariant bilinear form \(f\) they similarly investigate the decomposition of \(\mathcal T\) as the direct sum of \(f\)-indecomposable ideals; i.e. ideals that do not contain proper nonzero ideals \(\mathcal I\) with \(\mathcal I \cap \mathcal I^\perp = 0\). The final section contains applications to Lie triple systems.