Quadratic four-direction box spline ($\Xi =(e_1,e_2,e_1+e_2,e_1-e_2)$)

This box spline is the density of the image of the uniform unit four-cube under the indicated direction matrix. Its polynomial pieces have total degree $4-2=2$. Removing three directions is necessary to leave a nonspanning set, giving $C^1$ continuity. Its knot arrangement is a criss-cross triangulation of a square lattice. Its refinement mask is $\frac{1}{4}(1+z)(1+w)(1+zw)(1+z/w)$, up to the monomial that fixes the origin.