# Lie group structures inside of Clifford algebras

I am reading a text by Jean Gallier on Clifford algebras, Pin and Spin groups. I have a problem with one little innocent-looking paragraph establishing Pin and Spin as Lie groups on the page 37. I don't know what he uses in the arguments. I'm gonna go through the difficulties I have and add some of my reasoning. I would be glad for any light shed on them.

What is the topology? Let $C_{p,q}$ denote the real Clifford algebra associated with a non-degenerate quadratic form of signature $(p,q)$. Its underlying vector space $V$ is isomorphic to a certain Euclidean space ($V \cong \mathbb{R}^{2^{p+q}}$). I assume, the euclidean topology from this isomorphism is THE topology for $C_{p,q}$, but I'm not really sure.

Why is the group of units a topological group? How do we know, that multiplication and inverses are even continuous? Since the multiplication is inherited from tensor algebra, I don't see how it all plays together.

Continuity of the map $C_{p,q}\rightarrow \mathrm{End}(V)$ given by $x \mapsto (y\mapsto xy)$. Maybe If I understood the multiplication (previous point), I would see how is this continuous.

If I would understood these, then it would be clear, why the group of units $C_{p,q}^\times$ is an open subset. But why is it a Lie group? It is certainly locally euclidean, hence by a fundamental theorem of Gleason, Montgomery and Zippin, it is a Lie group. Is there an easier way to see that? I guess the author would at least mentioned that he uses such a big gun. What we effectively need is just to show that multiplication is smooth. Am I right? I'm rather unfamiliar with the Lie theory.

Then Pin and Spin groups are closed subgroups of a Lie group and hence both Lie groups themselves. The fact, that they are given by closed condition isn't crystal clear to me - but I believe if I've understood the multiplication's continuity, I would be able to supply a formal proof of continuity of the norm.

I'm sorry that this is a somewhat open question. Can anyone direct me from the abyss of Lie ignorance?