An analogue of a result of Tits for linear and symplectic transvection groups

In [9] Bogdan Nica presented an elementary proof of a result which says that the relative elementary linear group with respect to a square of an ideal of the ring is a subset of the true relative elementary linear group. The original result was proved by J. Tits in [16] in the much general context of Chevalley groups. In this paper we prove analogues of the result of Tits for linear transvection group and symplectic transvection group. We also obtain an elementary proof of a special case of the result by Tits, namely the case of elementary symplectic group, using commutator identities for generators of this group.