  Following Craven and Rouquier's computational method to tackle Brou\'e's
abelian defect group conjecture, we present two algorithms implementing that
procedure in the case of principal blocks of defect $D \cong C_{\ell} \times
C_{\ell}$ for a prime $\ell$; the first describes a stable equivalence between
$B_0(G)$ and $B_0(N_G(D))$, and the second tries to lift a such stable
equivalence to a perverse derived equivalence. As an application, we show that
Brou\'e's conjecture holds for the principal $5$-block of the simple group
$\Omega^{+}_8(2)$.
