With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: Show that for all $X \in \mathcal{M}_{2n,1}(\mathbb{R})$, $J_{n} X \in X^{\perp}$ and compute $\varphi(J_{n} X, X)$.
With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: Show that for all $X \in \mathcal{M}_{2n,1}(\mathbb{R})$, $J_{n} X \in X^{\perp}$ and compute $\varphi(J_{n} X, X)$.