Q: Complexification of real irreducible representations I have been reading this set of notes: http://www.math.columbia.edu/~rf/realreps.pdf
In section 4, theorem 4.1, there is a classification of the structure of $V$, the complexification of $V_{\mathbb{R}}$ based on the $G$-endomorphisms of $V_{\mathbb{R}}$ as follows:
Let $V_{\mathbb{R}}$ be an irreducible real representation and let $V$ be its complexification.
\begin{array}{h}
(\text{i}) & \text{Hom}^{G}(V_{\mathbb{R}}, V_{\mathbb{R}})\cong\mathbb{R}\iff V \text{ is irreducible.} \\
(\text{ii}) & \text{Hom}^{G}(V_{\mathbb{R}}, V_{\mathbb{R}})\cong\mathbb{C}\iff V\cong W\oplus W^{\ast}\text{ where $W$ and $W^{\ast}$ are irreducible and}\\&\text{ not isomorphic.} \\
(\text{iii}) & \text{Hom}^{G}(V_{\mathbb{R}}, V_{\mathbb{R}})\cong\mathbb{H}\iff V\cong W\oplus W,\text{ where $W$ is irreducible and $W\cong W^{\ast}$}.
\end{array}
The proof is omitted, so I am trying to prove it myself, but don't really know where to go with this. All that I have so far is that End$(V)\cong\text{End}(V_{\mathbb{R}})\otimes_{\mathbb{R}}\mathbb{C}$, as I thought this would be useful.
Any pointers would be much appreciated. Thanks!

A: Here is a sequence of exercises. The notation will be more uniform if we switch it to the following: I'll write $V$ for the real representation and $V_{\mathbb{C}} \triangleq V \otimes_{\mathbb{R}} \mathbb{C}$ for its complexification.

*

*Using Schur's lemma, prove that $\text{End}_G(V)$ is a finite-dimensional (associative) division algebra over $\mathbb{R}$.


*Prove that the only finite-dimensional (associative) division algebras over $\mathbb{R}$ are $\mathbb{R}, \mathbb{C}$, and $\mathbb{H}$. This is the Frobenius theorem and it's a bit tricky so you can skip this and just assume it is true for simplicity if you want.


*Prove that $\text{End}_G(V_{\mathbb{C}}) \cong \text{End}_G(V)_{\mathbb{C}}$ (see how nice the notation is this way?), which you've already done. Compute the complexifications of $\mathbb{R}, \mathbb{C}, \mathbb{H}$.


*Prove that in the direct sum decomposition of $V_{\mathbb{C}}$, if $W$ is an irreducible representation over $\mathbb{C}$ then the multiplicity of $W$ and of $W^{\ast}$ must be the same. (It's a bit misleading to work with the dual here; really we are working with the complex conjugate $\overline{W}$, which is the same representation but with scalar multiplication modified by a conjugate. It just happens that we always have $W^{\ast} \cong \overline{W}$.)


*Write $V_{\mathbb{C}}$ as a direct sum of irreducible representations and compute $\text{End}_G(V_{\mathbb{C}})$ in terms of it. Now apply exercises 3 and 4; what do you conclude about the irreducible decomposition of $V_{\mathbb{C}}$?
