Summary: We review various aspects of cluster algebras and the ways in which they appear in the study of loop-level amplitudes in planar \( \mathcal{N}=4 \) supersymmetric Yang-Mills theory. In particular, we highlight the different forms of cluster-algebraic structure that appear in this theory’s two-loop MHV amplitudes – considered as functions, symbols, and at the level of their Lie cobracket – and recount how the ‘nonclassical’ part of these amplitudes can be decomposed into specific functions evaluated on the \(A_2\) or \(A_3\) subalgebras of Gr\((4,n)\). We then extend this line of inquiry by searching for other subalgebras over which these amplitudes can be decomposed. We focus on the case of seven-particle kinematics, where we show that the nonclassical part of the two-loop MHV amplitude is also constructible out of functions evaluated on the \(D_5\) and \(A_5\) subalgebras of Gr(4,7), and that these decompositions are themselves decomposable in terms of the same \(A_4\) function. These nested decompositions take an especially canonical form, which is dictated in each case by constraints arising from the automorphism group of the parent algebra.