Abstract: Building on work of the first and last author, we prove that an embedding of simple affine vertex algebras , corresponding to an embedding of a maximal equal rank reductive subalgebra into a simple Lie algebra , 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 decomposes finitely as a -module.

Communicated by Y. Kawahigashi. D.A. and O.P. are partially supported by the Croatian Science Foundation under the project 2634 and by the Croatian Scientific Centre of Excellence QuantixLie.