TITLE: Lie algebras with unique invariant scalar product
QUESTION [6 upvotes]: Every 1-dimensional or simple complex Lie algebra admits an invariant, symmetric and non-degenerate bilinear form. This form is unique up to multiplication by a nonzero constant (which in Yang-Mills theories plays the role of the coupling constant). 
Are there other Lie algebras with such a unique scalar product?

REPLY [3 votes]: I. Bajo and S. Benayadi already gave a complete answer for your question in their paper entitled "Lie algebras admitting a unique quadratic structure" in 1997, there you can find much more detail.