Commutators in pseudo-orthogonal groups

We study commutators in pseudo-orthogonal groups O2n R (including unitary, symplectic, and ordinary orthogonal groups) and in the conformal pseudo-orthogonal groups GO2nR. We estimate the number of commutators, c(O2nR) and c(GO2nR), needed to represent every element in the commutator subgroup. We show that c(O2nR)≤4 if R satisfies the A-stable condition and either n≥ 3 or n = 2 and 1 is the sum of two units in R, and that c(GO2nR)≤3 when the involution is trivial and A = R ∊. We also show that c(O2nR)≤3 and c(GO2nR)≤ 2 for the ordinary orthogonal group O2n R over a commutative ring R of absolute stable rank 1 where either n > 3 or n = 2 and 1 is the sum of two units in R.