Commutation relations for quantum root vectors of cominuscole parabolics

We prove a result for the commutator of quantum root vectors corresponding to cominuscole parabolics. Specifically we show that, given two quantum root vectors, belonging respectively to the quantized nilradical and the quantized opposite nilradical, their commutator belongs to the quantized Levi factor. This generalizes the classical result for Lie algebras. Recall that the quantum root vectors depend on the reduced decomposition of the longest word of the Weyl group. We show that this result does not hold for all such choices. We conjecture that it holds when the reduced decomposition is appropriately factorized.