Definitions of a geometric algebra/exterior algebra

Question

I'm interested in exterior algebra and similar topics and I'm reading this article: https://en.wikipedia.org/wiki/Geometric_algebra#Definition_and_notation I've got two questions:

1. Do you assess this definition is formally correct? In my opinion: it's not. In particular $\mathcal{G}(p,q)$ in $A, B, C\in \mathcal{G}(p,q)$ is another $\mathcal{G}(p,q)$ than $\mathcal{G}(p,q)$ in $AB \in \mathcal{G}(p,q)$ isn't it? But the "definition" doesn't explain what set $AB$ belongs to and hence it loops. The definition also doesn't say what the codomain of $g$.
2. Where the second equation in $$\frac{1}{2}(ab + ba) = \frac{1}{2}\left((a + b)^2 - a^2 - b^2\right) = a \cdot b$$ comes from?

Answer 1

1. a) Do you assess the definition is formally correct?

Yes, it's fine. It just says "the exterior algebra is the Clifford algebra with the zero bilinear form."

1. b) In particular $\mathcal{G}(p,q)$ in $A, B, C\in \mathcal{G}(p,q)$ is another $\mathcal{G}(p,q)$ than $\mathcal{G}(p,q)$ in $AB \in \mathcal{G}(p,q)$ isn't it? But the "definition" doesn't explain what set $$ belongs to and hence it loops.

I have no idea what that means. The only place I see $AB$ is in the list of properties (I hope you didn't take that to be a definition. It seems you may have.) And that one place seems to say exactly what set $AB$ belongs to, and you seem to have typed it no less than 5 times above... I don't think any looping is involved because there isn't really any defining going on at that point.

1. c) The definition also doesn't say what the codomain of $$.

It's true it doesn't explicitly say it here, but a "bilinear form $g$ on $F$ vector space $V$" means a function $g:V\times V\to F$ with special properties.

2. Where the second equation in [...] comes from? [sic]

For vectors $a,b$ and a bilinear form $B$ for a field that's not characteristic $2$, there is always a way to separate $B$ into symmetric and antisymmetric pieces. That's what that equation is doing. Actually, this is something that is done more generally for tensors, and a bilinear form is just an example of a $2$ tensor.

In this particular case you can just verify it using basic algebra. There is not much to say about the equation.