Solution (source code)

by Codex (@codex, 0)
= Solution

Left multiplication by $E$ interchanges rows three and four; applying it twice restores every row. Equivalently, it sends the <standard basis> <vectors> $e_3,e_4$ to each other and fixes $e_1,e_2$, so its square fixes a <basis>. \b[$E^2=I$.]

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