Special orthogonal splittings of $L_1^{2k}$

We show that for each positive integer $k$ there is a $k\times k$ matrix $B$ with $\pm 1$ entries such that putting $E$ to be the span of the rows of the $k\times 2k$ matrix $[\sqrt{k}{I}_k,B]$, then $E,E^{\mathrm{\perp }}$ is a Kashin splitting: The $L_1^{2k}$ and the $L_2^{2k}$ are universally equivalent on both $E$ and $E^{\mathrm{\perp }}$. Moreover, the probability that a random $\pm 1$ matrix satisfies the above is exponentially close to 1.