In Howard Georgi's book Lie Algebras In Particle Physics, it is claimed in Problem 12.C. that for any Lie group, the tensor product of the adjoint representation with any arbitrary nontrivial representation D must contain D.
I cannot figure out why it is so. I found a similar question For the adjoint and any nontrivial representation , why is in ⊗? But I don't understand the last part It follows ... in the answer.
The categories mentioned in that example are semi-simple, implying that every surjection $M \to N$ has a right inverse, so that $N$ is isomorphic to a subobject of $M$. For a compact Lie group, its category of finite-dimensional representations is semi-simple thanks to the fact that each such representation carries an invariant positive-definite Hermitian form (which may be obtained by averaging over the group, just as in the case of finite groups).