This is some sort of "follow-up" to the (unanswered) question posted here.

Let's denote $$\varphi :O(2n)\rightarrow O(2n);A\mapsto det(A)\cdot A.$$ Then $\varphi $ is an automorphism of $O(2n)$, and as is shown in the above link, when $n$ is even, $$H^2(B\varphi;Z/2): H^2(BO(4k);Z/2)\rightarrow H^2(BO(4k);Z/2)$$ switches the classes $w_2$ and $w_2+w_1^2$. Since the extension $$Z/2\rightarrow Pin^{-}(4k)\rightarrow O(4k) $$ is classified by the class $w_2$ and $$Z/2\rightarrow Pin^{+}(4k)\rightarrow O(4k) $$ is classified by the class $w_2+w_1^2$ c.f. introduction of Kirby, Taylor, "A Calculation of Pin^{+}bordism groupos" Comment. Math. Helvetici 65 (1990) 434-447, we conclude that $\varphi$ pulls back the one extension to the other. In particular, as Lie groups $Pin^+(4k) $ and $Pin^-(4k) $ are isomorphic.

Now my questions are :

- Is this isomorphism documented somewhere?
- Is there a simple way to see this directly from the definition of $Pin$ groups using Clifford algebras?