Finite vs infinite decompositions in conformal embeddings

Building on work of the first and last author, we prove that an embedding of simple affine vertex algebras $V_{mathbf{k}}(g^0)subset V_{k}(g)$, corresponding to an embedding of a maximal equal rank reductive subalgebra $g^0$ into a simple Lie algebra $g$, is conformal if and only if the corresponding central charges are equal. We classify the equal rank conformal embeddings. Furthermore we describe, in almost all cases, when $V_{k}(g)$ decomposes finitely as a $V_{mathbf{k}}(g^0)$-module.