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What is the normalizer of SU(2) x SU(2) in SU(4) or how would I find it?

Reason for the question: with 2 qubits, if I was interested in conjugation of 2-qubit gates with generic SU(2) elements, instead of with elements of the Pauli group, what 2-qubit gates would be preserved by such operation (similarly to Clifford gates for the case of Pauli group)?

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    $\begingroup$ This group preserves exactly 2 subspaces, so the normalizer preserved this (unordered) pair of subspaces. This describes the normalizer in $U(4)$ as the obvious overgroup of index 2 of $U(2)\times U(2)$. This is certainly not research-level. $\endgroup$
    – YCor
    Commented Nov 26, 2021 at 17:22
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    $\begingroup$ The normalizer is those matrices with block form $\begin{bmatrix} u&0\\0&v \end{bmatrix}$ or $\begin{bmatrix} 0&u\\v&0 \end{bmatrix}$ with $u$ and $v \in U(2)$ and with $\det(u) = \det(v)^{-1}$. $\endgroup$
    – David E Speyer
    Commented Nov 26, 2021 at 17:23

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