$\DeclareMathOperator\Cl{Cl}$It is known that every Clifford Algebra $\Cl(Q)$ over the real numbers where $Q: \mathbb R^n \to \mathbb R$ is a non-degenerate quadratic form is isomorphic to a matrix algebra over $\{\mathbb R, \mathbb C, \mathbb H, \mathbb R \oplus \mathbb R, \mathbb C \oplus \mathbb C, \mathbb H \oplus \mathbb H\}$. What I'm wondering is whether, for any such $Q$, there is an isomorphism so that the generators of the Clifford algebra are represented as normal matrices. In order to define a normal matrix, we need to define an involution for the above six algebras: For $\mathbb R$, we define $z^* = z$; for $\mathbb C$, we define $(a + bi)^* = a - bi$; for $\mathbb H$, we define $(a + bi + cj + dk)^* = a - bi - cj - dk$; for $R \oplus R$ for any $*$-ring $R$, we define $(w,z)^* = (w^*,z^*)$.
I know that the so-called Dirac representation for $\Cl_{n,n}(\mathbb R)$ satisfies my criteria. But I would like a reference for a "normal representation" for $\Cl_{p,q}(\mathbb R)$ for any $p$ and $q$. I can work this out myself, but it's tedious, and I'd like to see if there's a reference.
Alternative phrasing
An alternative way of phrasing the question is that we want to strengthen the ring isomorphisms in the classification theorem for Clifford algebras into $*$-ring isomorphisms. Details:
Here is the Wikipedia article on $*$-rings.
$\Cl_{p,q}(\mathbb R)$ can be made into a $*$-ring by defining $e^* = \begin{cases} -e, &\text{if } e^2 = -1 \\ e, &\text{if } e^2 = 1\end{cases}$ for any generator $e$, and extending to $\Cl_{p,q}(\mathbb R)$ by linearity.
Likewise, $A \in M_n(R)$ (where $R$ is some $*$-ring) can have $A^*$ defined as $(A^*)_{ij} = A_{ji}^*$.
This then turns $M_n(R)$ into a $*$-ring where $R$ is any of $\{\mathbb R, \mathbb C, \mathbb H, \mathbb R \oplus \mathbb R, \mathbb C \oplus \mathbb C, \mathbb H \oplus \mathbb H\}$.
The question is, does there exist a $*$-ring isomorphism between $\Cl_{p,q}(\mathbb R)$ and some $M_n(R)$, as opposed to merely a ring isomorphism?
