  In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct
splittings of $bo \wedge bo$ and $BP\langle 1 \rangle \wedge BP\langle 1
\rangle$. These splittings helped make it feasible to do computations using the
$bo$- and $BP \langle 1 \rangle$-based Adams spectral sequences. In this paper,
we construct an analogous splitting for $BP \langle 2 \rangle \wedge BP\langle
2 \rangle$.
