I think the Jussieu webpages moved at some point but this link should be now correct and there is also the github source which should be a more permanent link.

That is an error in the book of Geck & Pfeiffer that comes from some subtle labelling issues. For the correction see Lemma 3.5 in the paper "On Kottwitz' conjecture for twisted involutions" by Geck arxiv.org/abs/1206.0443

Relatedly $e_i^*e_j^*$ is also not zero. Let $v = e_i + e_j$ then $e_i^*(v) = e_j^*(v) = 1$ and so $(e_i^*e_j^*)(v) = e_i^*(v)e_j^*(v) = 1\cdot 1 = 1$. Recall we evaluate the product pointwise. In fact, in the example in the previous comment the quadratic form is $q = e_1^*e_2^*$.

That doesn't imply $q_{rs} = 0$ because $q_{rs}$ is not linear. Take $n=2$ and let $\beta$ be the symmetric bilinear form with $\beta(e_1,e_2) = \beta(e_2,e_1) = 1$ and $\beta(e_1,e_1) = \beta(e_2,e_2) = 0$. We have a quadratic form given by $q(v) = \frac{1}{2}\beta(v,v)$ for all $v \in V$. Certainly $q(e_1) = q(e_2) = 0$ but $q(e_1+e_2) = \frac{1}{2}\beta(e_1+e_2,e_1+e_2) = 1 \neq 0$. So it is not enough to check this on a basis.

For a textbook reference you could look at Theorem 4.1.12 of Geck's "An Introduction to Algebraic Geometry and Algebraic Groups". This is basically the proof given by David Speyer below but it is written up nicely.

@LSpice Glad it worked, otherwise I would have had to shamefully admit that it's a link to one of my own papers.

$X/\mathbb{Z}\Phi = X(Z(G))$ and $X/L = X(Z^{\circ}(G))$ (scheme theoretically). So your decomposition recovers $(G/Z^{\circ}(G),Z^{\circ}(G))$. Which makes sense as you should be able to recover $Z(G)/Z^{\circ}(G)$ from $X$ and $\Phi$.

Just to say, your viewpoint of the product map $G_{\mathrm{der}} \times Z^{\circ}(G) \to G$ is essentially the point of view taken in this paper: doi.org/10.1017/S0013091518000597

If $W$ is a finite Weyl group then it is known that for $w \in W$ we have $\mathbf{a}(w) = \mathbf{a}_{\lambda}$ for some complex irreducible character $\lambda \in \mathrm{Irr}(W)$, see Prop. 2.3.14 of the book by Geck–Jacon. The values $a_{\lambda}$ have all been computed by Lusztig (naturally) and are contained in the CHEVIE software maintained by Jean Michel. See also Chapter 6 of the book by Geck–Pfeiffer. There combinatorial descriptions are given for $\mathbf{a}_{\lambda}$ when $W$ is classical.