The coefficients aα are determined from the commutation relations as follows: N α,βaα+β = Nθ[α],θ[β]aαaβ. Moreover, because θ is an automorphism of the root lattice we have N2 = N 2 , α,β θ[α],θ[β] and so if aα and aβ are equal to ±1, then so is aα+β. But since this is true for the simple roots it remains true for all roots.