Let $L$ be a real vector space of dimension 22 and $q$ a quadratic form on $L$ of signature $(3,19)$.

Let $V\subset L$ be a positive oriented subspace of dimension 2 and $G^{po}(2,L)$ be the Grassmannian of positive and oriented planes in $L$. I have read that the tangent space of $G^{po}(2,L)$ in $V$ is canonically identified with $Hom(V,V^\perp)$ (the orthogonality is intended with respect to $q$, of course).

I can not find a way to view this. I know that $Gr^o(2,L)$, the Grassmannian of oriented planes in $L$, is a double cover of $Gr(2,L)$ and is locally an isometry, so this two spaces have the same tangent spaces. Besides, i think there is no problem identifying $Hom(V,L/V)$ with $Hom(V,V^\perp)$. So my question now is: how does the positivity not change the tangent spaces?