Killing form - strange definition

Question

I was just reading about Killing forms. In my opinion, the definition of these forms is quite strange. I mean why would one define $B(X,Y) = \mathrm{tr} (\mathrm{ad} (X) \circ \mathrm{ad} (Y))$? I would have expected $B(X,Y) = \mathrm{tr} (\mathrm{ad} (X)^* \circ \mathrm{ad} (Y)).$ At least for matrix groups, this seems to be a more intuitive definition, similar to the standard Frobenius inner product for matrices.

Answer 1

"Why would one define $B(X,Y)=tr(ad(X)ad(Y))$ ?" You are right, for matrix algebras I would define a bilinear form more simply, namely just by $C(X,Y)=tr(X)tr(Y)$. This is very natural, because a trace form for linear operators is the easiest thing you can imagine. If we do not have linear operators $X,Y$, then we can enforce this, by using the adjoint representation of the Lie algebra. So it is very natural, to use the trace of $ad(X)$ and $ad(Y)$ (and it has many other advantages, as mentioned in the comments).
Later it turns out that for simple Lie algebras, both forms are almost identical. They just differ by a non-zero constant, i.e. $B(X,Y)=\lambda C(X,Y)$.