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In the 1980s, Mahowald and Kane used Brown-Gitler spectra to constructsplittings 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 $BP \langle 2 \rangle \wedge BP\langle2 \rangle$.

On a Spectrum-level Splitting of the $BP \langle 2 \rangle$-Cooperations Algebra