U(m) as a subgroup of SO(2m)

Question

We know $U(m)$ is one the subgroups of $SO(2m)$ acting transitively on the sphere $S^{2m-1}$ (one of the groups in the Borel's list). What is the explicit formula of this embedding (or it's action)?

Answer 1

$U(m)$ acts naturally on the unit sphere in $\mathbf C^m$ (vectors of norm $1$ for the standard Hermitian form), which under the identification of $\mathbf C$ with $\mathbf R^2$ becomes the unit sphere $\mathbf S^{2m-1}$. So the embdedding is just restriction of scalars from $\mathbf C$-linear operators to $\mathbf R$-linear operators.