The Howe-Moore theorem implies that any ergodic action of a connected noncompact, finite-center, simple Lie group is mixing. Thus, for example, ergodicity inherits to all noncompact closed subgroups, a very useful fact which immediately yields ergodicity of many geometrically-motivated actions. The proofs I know of Howe-Moore go via unitary representation theory. By thinking of a Hilbert space as an infinite-dimensional Riemannian manifold and by adjusting techniques originally used in studying Lorentzian (and Riemannian) dynamics, we can obtain a version of Howe-Moore that is valid for all connected Lie groups, though only useful for nonAbelian groups. Specifically, for any connected Lie group \$G\$, for any faithful irreducible unitary representation of \$G\$, we have: any matrix coefficient tends to zero, as \$Ad(g)\$ leaves compact subsets of \$GL({\frak g})\$, where \${\frak g}\$ is the Lie algebra of \$G