Invariant Non degenerate symmetric bilinear forms on semisimple lie algebras?

Question

We know every finite-dimensional semisimple lie algebra can be written as direct sum of simple lie algebras. Also, everybody knows all invariant symmetric bilinear and non degenerate forms on simple lie algebra is rescaling of killing form.

Is it true for semisimple ones?

Answer 1

For semisimple Lie algebras $\mathfrak{s}=\mathfrak{s}_1\oplus \cdots \mathfrak{s}_r$ one can rescale the invariant symmetric non-degenerate bilinear form on each simple summand $\mathfrak{s}_i$ independently. Hence the bilinear form on $\mathfrak{s}$ need not be a multiple of the Killing form.