  Using tensor categories, we present new constructions of several of the
exceptional simple Lie superalgebras with integer Cartan matrix in
characteristic $p = 3$ and $p = 5$ from the complete classification of modular
Lie superalgebras with indecomposable Cartan matrix and their simple
subquotients over algebraically closed fields by Bouarroudj, Grozman, and
Leites in 2009. Specifically, let $\mathbf{\alpha}_p$ denote the kernel of the
Frobenius endomorphism on the additive group scheme $\mathbb{G}_a$ over an
algebraically closed field of characteristic $p$. The Verlinde category
$\mathrm{Ver}_p$ is the semisimplification of the representation category
$\mathrm{Rep} \ \mathbf{\alpha}_p$, and $\mathrm{Ver}_p$ contains the category
of super vector spaces as a full subcategory. Each exceptional Lie superalgebra
we construct is realized as the image of an exceptional Lie algebra equipped
with a nilpotent derivation of order at most $p$ under the semisimplification
functor from $\mathrm{Rep} \ \mathbf{\alpha}_p$ to $\mathrm{Ver}_p$.
