Let $x,y,v$ be vectors with $v\cdot v=1$.
Then we have that \begin{align*}\left [x-2(x\cdot v)v\right ]\cdot \left [y-2(y\cdot v)v\right ]&=x\cdot y-2(x\cdot v)(y\cdot v)-2(x\cdot v)(y\cdot v)+4(x\cdot v)(y\cdot v)(v\cdot v) \\ & = x\cdot y-4(x\cdot v)(y\cdot v)+4(x\cdot v)(y\cdot v) \\ & = x\cdot y\end{align*} Would it be correct to do the following instead? \begin{align*}\left [x-2(x\cdot v)v\right ]\cdot \left [y-2(y\cdot v)v\right ]&=\left (x-2x\cdot v^2\right )\cdot \left (y-2y\cdot v^2\right )\\ &=\left (x-2x\right )\cdot \left (y-2y\right ) \\ &=\left (-x\right )\cdot \left (-y\right ) \\ & = x\cdot y\end{align*}