Separable equivalence of rings and symmetric algebras

We continue a study of separable equivalence from (Kadison, Comptes Rendus Math. Reports Acad. Sci. Canada, 15 (1993) 223-228; Hokkaido Math. J. 24 (1995) 527-549). We prove that symmetric separable equivalent rings A and B are linked by a Frobenius bimodule APB such that A is P-separable over B. Separably equivalent rings are linked by a biseparable bimodule P. In addition, the ring monomorphism AEndPB is split, separable Frobenius. It is observed that left and right finite projective bimodules over symmetric algebras are Frobenius bimodules; twisted by the Nakayama automorphisms if over Frobenius algebras.