  Let $V$ be a finite-dimensional $FG$-module, where $F$ is a field of prime
characteristic $p$ and $G$ is a group. We show that, when $r$ is not a power of
$p$, the Lie power $L^r(V)$ has a direct summand $B^r(V)$ which is a direct
summand of the tensor power $V^{\otimes r}$ and which satisfies $\dim
B^r(V)/\dim L^r(V) \to 1$ as $r \to \infty$. Similarly, for the same values of
$r$, we obtain a projective submodule $C(r)$ of the Lie module $\Lie(r)$ over
$F$ such that $\dim C(r)/\dim \Lie(r) \to 1$ as $r \to \infty$.
