Proposition 1

Let A be an $\mathbf{R}$-algebra (associative and unital) and V a finite dimensional simple A-module over $\mathbf{R}$. Then, we must be in one of the following three situations:

$\alpha )$ The commutant of V (Algebra, Chap. VIII, § 5, no. 1) is isomorphic to $\mathbf{R}$, and the $A_{(\mathbf{C})}$-module $V_{(\mathbf{C})}$ is simple;

$\beta )$ the commutant of V is isomorphic to $\mathbf{C}$; the $A_{(\mathbf{C})}$-module $V_{(\mathbf{C})}$ is the direct sum of two non-isomorphic simple $A_{(\mathbf{C})}$-submodules which are interchanged by $\sigma \otimes 1_V$;

$\gamma )$ the commutant of V is isomorphic to $\mathbf{H}$; the $A_{(\mathbf{C})}$-module $V_{(\mathbf{C})}$ is the direct sum of two isomorphic simple $A_{(\mathbf{C})}$-submodules that are interchanged by $\sigma \otimes 1_V$.

The commutant E of V is a field, a finite extension of $\mathbf{R}($Algebra, Chap. VIII, §3, no. 2, Prop. 2), hence isomorphic to $\mathbf{R},\mathbf{C}$ or $\mathbf{H}($Algebra, Chap. VIII, §15). The $A_{(\mathbf{C})}$-module $V_{(\mathbf{C})}$ is semi-simple (Algebra, Chap. VIII, §11, no. 4), and its commutant can be identified with $\mathbf{C}\otimes_{\mathbf{R}}E ($Algebra, Chap. VIII, §11, no. 2, Lemma 1).

If E is isomorphic to $\mathbf{R}$, the commutant of $V_{(\mathbf{C})}$ is isomorphic to $\mathbf{C}$, and $V_{(\mathbf{C})}$ is a simple $A_{(\mathbf{C})}$-module (Algebra, Chap. VIII, §11, no. 4).

If E is not isomorphic to $\mathbf{R}$, it contains a field isomorphic to $\mathbf{C}$; it follows that V has an $A_{(\mathbf{C})}$-module structure, denoted by $V^c$. Then $V^c$ is a simple $A_{(\mathbf{C})}$-module, and the $\mathbf{C}$-linear map $\psi : V_{(\mathbf{C})}\rightarrow V^c\oplus \overline{V}^c$ such that $\psi (\alpha \otimes v) =$ $(\alpha v,\overline{\alpha v})$ for $\alpha \in \mathbf{C},v\in V$, is an isomorphism (Algebra, Chap. V, §10, no. 4, Prop. 8). Moreover, $\sigma \otimes 1_V$ corresponds under this isomorphism to the $\mathbf{R}$-automorphism $(v, v') \rightarrow (v', v)$ of $V^c\oplus \overline{V}^c$, and hence interchanges the two $A_{(\mathbf{C})}$-submodules $\psi^{-1}(V^c)$ and $\psi^{-1}(V^c)$.

The commutant $E_{(\mathbf{C})}$ of $V_{(\mathbf{C})}$ thus contains $\mathbf{C}\times \mathbf{C}$, operating by homotheties on $V^c\oplus \overline{V}^c$. There is no isomorphism of $A_{(\mathbf{C})}$-modules from $V^c$ to $\overline{V}^c$ if and only if $E_{(\mathbf{C})}$ reduces to $\mathbf{C}\times \mathbf{C}$, that is, if E is isomorphic to $\mathbf{C}$. This completes the proof.