  Let $\mathfrak{r}$ be a finite dimensional complex Lie superalgebra with a
non-degenerate super-symmetric invariant bilinear form, let $\mathfrak{p}$ be a
finite dimensional complex super vector space with a non-degenerate
super-symmetric bilinear form, and let $\nu:
\mathfrak{r}\rightarrow\mathfrak{osp}(\mathfrak{p})$ be a Lie superalgebra
homomorphism. In this paper, we give a necessary and sufficient condition for
$\mathfrak{r}\oplus\mathfrak{p}$ to be a quadratic Lie superalgebra. The
criterion obtained is an analogue of a constancy condition given by Kostant in
the Lie algebra setting. As an application, we prove an analogue of the
Parthasarathy's formula for the square of the Dirac operator attached to a pair
of quadratic Lie superalgebras.
