Invariant symmetric bilinear forms on Lie triple systems

We first show that an invariant symmetric bilinear form on a Lie triple system can be uniquely extended to its standard imbedding Lie algebra, and then prove that any Lie triple system over a field F of characteristic zero admitting a unique up to a scalar multiple, nondegenerate invariant symmetric bilinear form is necessarily simple. If the field F is algebraically closed, that condition is also sufficent.