Given a complex Hilbert space $\mathcal{H}$ of dimension $d$ interpreted as vectorspace over $\mathbb{R}$ with dimension $2d$ and the space $L(\mathcal{H},\mathbb{C}^4)$ of all linear operators from $\mathcal{H}$ to $\mathbb{C}^4$ also interpreted as vectorspace over $\mathbb{R}$. Does there exists any subspace $E \subset L(\mathcal{H},\mathbb{C}^4)$ such that for any $B\in E$ and all $U \in U(\mathbb{C}^4)\setminus \{\pm I\}$ : $UB \notin E$?
In other words I'm looking for a real-subspace $E$ of dimension $\dim_\mathbb{R}(E) = 8d-16$ such that the map
$$R: E\subset L(\mathcal{H},\mathbb{C}^4) \to L(\mathcal{H})$$ $$ B \mapsto B^*B$$
is injective.