  We prove that a finite index regular inclusion of $II_1$-factors with
commutative first relative commutant is always a crossed product subfactor with
respect to a minimal action of a biconnected weak Kac algebra. Prior to this,
we prove that every finite index inclusion of $II_1$-factors which is of depth
$2$ and has simple first relative commutant (respectively, is regular and has
commutative or simple first relative commutant) admits a two-sided Pimsner-Popa
basis (respectively, a unitary orthonormal basis)
