There is a corollary to Schur's lemma which says that : If $V$ is a finite dimensional irreducible complex representation of a group G or Lie algebra and $\phi :V \rightarrow V$ is an intertwining map, then $\phi=\lambda I$ for some $\lambda \in \mathbb{C}$. In the proof that I know, we use the fact that the $V$ is a complex vector space and not just a real one.
1.) Please give a counterexample to the above if V is real.
I am reading Brian Hall's book on representation theory of Lie groups & Lie algebras and wish to settle this question in context of Problem 16, Ch.4.
Suppose that $V$ is an irreducible finite dimensional representation of a group or Lie algebra and consider the associated representation $ V \bigoplus V $. Show that every nontrivial irreducible invariant subspace of $ V \bigoplus V $ is of the type $ \{ (\lambda_1v,\lambda_2v)| v\in V\}$ for some constants $\lambda_1, \lambda_2$ not both zero.
My second question is
2.) Do we need to assume in this problem that $V$ is complex or is it true for real vector space representations as well ? If it is valid for real vector space representations, how does one prove it ?