norm condition imply determinant 1

Question

Suppose $(R^2,\|\cdot\|)$ be normed space and $x,y,\xi,\eta\in R^2$ such that $\|x\|=\|y\|=\|\xi\|=\|\eta\|=1$, $\|x+y\|=\|\xi+\eta\|=2$, $\|x-y\|=\|\xi-\eta\|$, $x,y$ and $\xi,\eta$ linearly independent respectively. Let $\xi=a_{11}x+a_{12}y$ and $\eta=a_{21}x+a_{22}y$. Is it true that $\left|\det \left(a_{ij}\right)\right|=1$? in other words $\left|a_{11}a_{22}-a_{12}a_{21}\right| = 1$?