Consider the Clifford algebra $\mathrm{Cl}(n)$ over Euclidean space $\mathbb{R}^n$ (with the standard inner product).
Now, in the case that $n$ is even, it is known (cf. [1]), then $\mathrm{Cl}(n)$ has a unique irreducible complex representation $\Delta$ (unique up to equivalence). Now, in [2], it is claimed (without proof) that this very fact implies that $$\mathrm{Cl}(n) \otimes_{\mathbb{R}} \mathbb{C} \simeq \mathrm{End}(\Delta), $$ and this is an isomorphism of algebras.
However, I cannot do find a proof or formulate a proof myself. I have a slight feeling that this might be either a trivial result, or a very well-known result, but I do not where to find a proof.
Thank you in advance.
[1] Spin Geometry, H. Blaine Lawson and Marie-Louise Michelsohn
[2] Elliptic operators, Topology, and Asymptotic methods, John Roe