I'm reading materials on spin groups. Let $Cl(s,t)$ be the Clifford algebra of $\mathbb{R}^{s+t}$ with standard bilinear form $\eta$ with signature $(s,t)$. The book then defines, $$\operatorname{Spin}^+(s,t)=\{v_1…v_{2p}w_1…w_{2q}\mid\eta(v_i,v_i)=1,\eta(w_j,w_j)=-1; \ p,q\geq 0\}$$ and said this forms a group. But, I don't see how can this be true? Since $v_i$ and $w_j$ are not necessarily be orthogonal with respect to $\eta$, we can not anti-commute $v_i$ and $w_j$, then how can a product like $w_1w_2v_1v_2$ be still in $\operatorname{Spin}^+(s,t)$?
EDIT: For those of you who met the same problem when reading the book Mathematical Gauge Theory by J.D. Hamilton, the following link contains a list of updates of the book including this point (350 (6)), it turns out this indeed it's a subgroup.
See EDIT in the question. It turns out that it's indeed a group, and a short proof is contained in the link attached, again, see edit.