A complex, finite dimensional Lie algebra admitting a (up to multiplication by a non-zero scalar) unique, invariant, symmetric, non-degenerate bilinear form is 1-dimensional or simple. Cf 'Lie algebras with unique invariant scalar product' June 20 2019.
I would like to drop the non-degeneracy:
Can you characterize all complex Lie algebras admitting a (up to multiplication) unique, invariant, symmetric bilinear form?
One easy example is the 2-dimensional Lie algebra [x,y]=y.