Let $q$ be a quadratic form of signature $(r,s+1)$. Then $Cl_{r,s+1}$ denotes the clifford algebra associated to $q$ over $R^{r+s+1}$ where $R$ is real number.
"$Cl_{r,s+1}\cong Cl_{s,r+1}$"
$\textbf{Q:}$ How do I see above isomorphism? Denote orthonormal basis of generators LHS as $e_1,\dots, e_r,f_1,\dots, f_{s+1}$ and RHS as $a_1,\dots, a_s,b_1,\dots, b_{r+1}$. It seems that I want to consider $e_i\to b_ib_{r+1}$ and $f_i\to a_ib_{r+1}$ in order to correct sign issue to allow $R^{r+s+1}\to Cl_{s,r+1}$ descend to $Cl_{r,s+1}$. However, I do not see how to send $f_{s+1}$? A choice would be $f_{s+1}\to b_{r+1}^2$ but that would give rise to injectivity issue.
Ref. Spin Geometry, Lawson Chpt 1, Sec 4.