This is likely a basic question, but I can't come up with a straightforward (dis)proof that the trace of generators of a Lie group are invariant. The reason I am asking is because the elements of the restricted Lorentz group have determinant $1$, and I wonder if the determinant is $1$ under other representations. I don't find this self-evident even though it is the case for the $(1/2,0), (0,1/2), (1/2,1/2)$ irreps.