  The space of based loops in $SL_n(\mathbb{C})$, also known as the affine
Grassmannian of $SL_n(\mathbb{C})$, admits an $\mathbb{E}_2$ or fusion product.
Work of Mitchell and Richter proves that this based loop space stably splits as
an infinite wedge sum. We prove that the Mitchell--Richter splitting is
coherently multiplicative, but not $\mathbb{E}_2$. Nonetheless, we show that
the splitting becomes $\mathbb{E}_2$ after base-change to complex cobordism.
Our proof of the $\mathbb{A}_\infty$ splitting involves on the one hand an
analysis of the multiplicative properties of Weiss calculus, and on the other a
use of Beilinson--Drinfeld Grassmannians to verify a conjecture of Mahowald and
Richter. Other results are obtained by explicit, obstruction-theoretic
computations.
