Separability in consistent truncations

The separability of the Hamilton-Jacobi equation has a well-known connection to existence Killing vectors and rank-two tensors. This paper combines this with detailed knowledge compactification metrics consistent truncations on spheres. fact that both inverse metric such compactifications, as well tensors can be written in terms bilinears underlying "round metric," enables us perform analyses for truncations. We introduce idea separating isometry show when truncation, without reduction gauge vectors, an isometry, then is always separable. When are present, group required abelian subgroup not impede separability. classify isometries spheres, $S^n$, $n=2, \dots, 7$, exhibit all corresponding These results may practical use identifying supergravity solutions belong generating separable amenable scalar probe calculations. Finally, while our primary focus equation, we also make some remarks about wave equation.