Is there a way to evaluate the trace of generators of the Lie algebra and group elements?
For example take $SO(N)$, with $\lbrace T^a\rbrace$ the set of generators, normalized such that $Tr(T^aT^b)=\delta^{ab}$ and $g= e^{i \theta^aT^a}$ is a group element of the Lie group.
Is there a way to compute the commutator of a Lie-algebra element with a Lie group element generated by the generators in the same representation? Something like $[T^a,g]=?$
( In fact I was wondering if there was a simple relation between the elements of the adjoint representation. If you compute the matrix of the action of $g$ on $T^b$ projected onto $T^a$ how is related to the projection of that same action g on $T^a$ projected onto $T^b$.)