The Hermitian modular group and the orthogonal group

The Hermitian modular group of degree $n$ over an imaginary quadratic field
$K=\mathbb{Q}(\sqrt{-m})$ was introduced by Hel Braun in the 1940s
as an analogue for the well known Siegel modular group. It acts on the
Hermitian half space and the accociated Hermitian modular forms have
been studied thoroughly in the past. However, this talk does not concentrate
on the modular forms but on the modular group itself. For $n=2$ and $m \neq 1,3$,
$m$ squarefree, we will prove that the Hermitian modular group $U(2,2;\mathcal{O}_K)$,
where $\mathcal{O}_K$ is the ring of integers in $K$,  is isomorphic to the discriminant
kernel of the orthogonal group O(2,4) and we will provide an explicit homomorphism.
Furthermore, we compute the normalizer of the Hermitian modular group in the symplectic
group and show that it is isomorphic to the integral orthogonal group which is the normalizer
of the discriminant kernel in the real orthogonal group.