  We investigate the consequences of elliptic leading singularities for the
unitarity-based representations of two-loop amplitudes in planar, maximally
supersymmetric Yang-Mills theory. We show that diagonalizing with respect to
these leading singularities ensures that the integrand basis is term-wise pure
(suitably generalized, to the elliptic multiple polylogarithms, as necessary).
We also investigate an alternative strategy based on diagonalizing a basis of
integrands on differential forms; this strategy, while neither term-wise
Yangian-invariant nor pure, offers several advantages in terms of complexity.
