  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 $E,E^{\bot}$ is a Kashin splitting:
The $L_1^{2k}$ and the $L_2^{2k}$ are universally equivalent on both $E$ and
$E^{\bot}$. Moreover, the probability that a random $\pm 1$ matrix satisfies
the above is exponentially close to 1.
