Commuting-scalar vacua of four-dimensional N=4 Yang-Mills theory (${\mathcal{M}}_{\mathrm{cl}}=({\mathbb{R}}^6{)}^r/\mathcal{W}$)

The scalar potential is a positive sum of squared commutators of the six adjoint scalar fields. It vanishes precisely when they commute. For a compact gauge group, simultaneous conjugation puts a commuting tuple in a Cartan subalgebra. Its arbitrary $6r$ real coordinates are classical flat directions of a scalar potential, with residual identification by the Weyl group. For $SU(N)$ the simultaneous eigenvalue vectors also obey the tracelessness constraint. Generic values break the gauge group to its maximal torus.