  In this paper we prove that the vertex algebra $V_L^+$ is rational if $L$ is
a negative definite even lattice of finite rank, or if $L$ is a non-degenerate
even lattice of a finite rank that is neither positive definite nor negative
definite. In particular, for such even lattices $L$, we show that the Zhu
algebras of the vertex algebras $V_L^+$ are semisimple. This extends the result
of Abe which establishes the rationality of $V_L^+$ when $L$ is a positive
definite even lattice of finite rank.
