I'm working with Clifford algebras, of which the first few are $C_0 = \mathbb{R}$, $C_1 = \mathbb{C}$, $C_2 = \mathbb{H}$, $C_3= \mathbb{H}^2$, $C_4 = M_{2,2}(\mathbb{H})$, $C_5= M_{4,4}(\mathbb{C})$, $C_6=M_{8,8}(\mathbb{R})$, $C_7 = M_{8,8}(\mathbb{R})^2$, $C_8 = M_{16,16}(\mathbb{R})$, etc. (The operations on the product of two algebras are componentwise.)
The notes I'm working with claim without proof that the minimum dimensions of representations are 1, 2, 4, 4, 8, 8, 8, 8, 16. (Sorry I probably could've formatted this better.) I assume that they mean faithful representations, and that they're representations as $\mathbb{R}$-algebras (and not just as rings), i.e. they're $\mathbb{R}$-linear maps $C_k \rightarrow End(V)$ (or equivalently maps $C_k \otimes_\mathbb{R} V \rightarrow V$, I think). I have no idea why these numbers are right, though. The first few I can see by ad hoc arguments, but that's not really that satisfying. I know no representation theory beyond a few definitions, so this might be really obvious and I just don't know the right theorems/arguments?