  Suppose $L$ is a semisimple Levi subgroup of a connected Lie group~$G$, $X$
is a Borel $G$-space with finite invariant measure, and $\alpha \colon X \times
G \to \GL_n(\real)$ is a Borel cocycle. Assume $L$ has finite center, and that
the real rank of every simple factor of~$L$ is at least two. We show that if
$L$ is ergodic on~$X$, and the restriction of~$\alpha$ to~$X \times L$ is
cohomologous to a homomorphism (modulo a compact group), then, after passing to
a finite cover of~$X$, the cocycle $\alpha$ itself is cohomologous to a
homomorphism (modulo a compact group).
