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

Rationality of the vertex algebra $V_L^+$ when $L$ is a nondegenerate even lattice of arbitrary rank