I am currently reading the book Spin Geometry by Lawson/Michelsohn to understand Dirac Operators and related topics. At some point it uses representation theory to classify Clifford Algebras. In particular this book states in I,§5. Theorem 5.6 on p.31:
Let $K=\mathbb{R}, \mathbb{C}$ or $\mathbb{H}$ and consider the ring $K(n)$ of $n \times n$ $K$-matrices as an algebra over $\mathbb{R}$. Then the natural representation $\rho$ of $K(n)$ on the vector space $K^n$ is, up to equivalence, the only irreducible representation of $K(n)$. The algebra $K(n) \oplus K(n)$ has exactly two equivalence classes of irreducible representations. They are given by $ \rho_1(\varphi_1,\varphi_2) = \rho(\varphi_1) \text{ and } \rho_2(\varphi_1,\varphi_2) =\rho(\varphi_2) $ acting on $K(n)$.
I am perfectly fine with the statement, but I would like to see a proof of that. Lawson/Michelsohn claim that this follows from the fact that the algebras $K(n)$ are simple and that simple algebras have only one irreducible representation up to equivalence. For the details the reader is referred to Lang: Algebra. Since this book as nearly 1000 pages I would like to ask, if someone could be a little more precise? Of course I would also be happy with a direct proof of the claim or a reference to any other readably textbook containing a proof. I must point out that I have no prior experience in the field of representation theory.