This similar post led me to this question. In the adjoint representation, I can construct the killing form to act as a metric on the elements in the algebra. It is often convenient to choose a basis so the metric (killing form) is the identity $\delta_{ij}$.
My question is if this choice enforces or induces a metric on any or all of the other representations (such as scaling by the dynkin index perhaps?), or if I am able to get away with having the identity metric in each representation.
