Lines & Planes in Space · Superchallenge problem

The regular tetrahedron has vertices $A = (1,1,1)$, $B = (1,-1,-1)$, $C = (-1,1,-1)$, $D = (-1,-1,1)$. Among all planes parallel to both edges $AB$ and $CD$ that cut through the tetrahedron, find the largest possible area of the cross-section.