# Universal gate sets and conjugates of $SU(2)$

Let $$G \subseteq SU(2)$$ be a finite set, and let $$E \in SU(2^k)$$ be any entangling gate (so $$k$$ is some number strictly greater than 1). We often say $$G$$ is universal iff $$\langle G \rangle$$ is dense in $$SU(2)$$, because then for any $$n$$, $$U \in SU(2^n)$$, and $$\epsilon > 0$$, there is an $$n$$-qubit circuit $$Q$$ over $$G \cup \{E\}$$ that approximates the output statistics of $$U$$ to within $$\epsilon$$ (w.r.t. total variation distance and computational basis inputs and measurements).

I am curious about a slight variation to this, which is more of a group theory question, but it's quantum-inspired nonetheless. Suppose $$G' \subseteq SL(2;\mathbb{C})$$ is a finite set, and let $$E \in SU(2^k)$$ be any entangling gate. If $$\langle G' \rangle$$ is dense in $$aSU(2)a^{-1}$$ for some $$a \in SL(2; \mathbb{C}) \backslash SU(2)$$, is it still the case that $$G' \cup \{E\}$$ is universal? That is, is it still the case that for any $$n$$, $$U \in SU(2^n)$$, and $$\epsilon > 0$$, there is an $$n$$-qubit circuit $$Q'$$ over $$G' \cup \{E\}$$ that approximates the output statistics of $$U$$ to within $$\epsilon$$ (again w.r.t. total variation distance and computational basis inputs and measurements)?