  Building on work of the first and last author, we prove that an embedding of
simple affine vertex algebras $V_{\mathbf{k}}(\mathfrak g^0)\subset
V_{k}(\mathfrak g)$, corresponding to an embedding of a maximal equal rank
reductive subalgebra $\mathfrak g^0$ into a simple Lie algebra $\mathfrak g$,
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 $V_{k}(\mathfrak g)$ decomposes finitely as a
$V_{\mathbf{k}}(\mathfrak g^0)$-module.
