In solving a physics problem, I came across a weird topological space constructed from $U(4)$, the group of $4\times4$ unitary matrices. I want to know the first two homotopy groups of it. Here is how it is defined:

Consider the set $X$ of matrices in $U(4)$ $$X=\{F\in U(4):F=N\Lambda\},$$ where $\Lambda$ is any diagonal unitary matrix, and $N$ has the block form $$N=\begin{bmatrix}A&A\\B&-B\end{bmatrix},$$ where $A$ and $B$ are $2\times2$ matrices. This, of course, means $N$ must be also unitary, and $A,B$ must satisfy $AA^\dagger=BB^\dagger=\frac{1}{2} I$. We can form the quotient space $M=X/\sim$, where $F_1\sim F_2$ if $F_1=F_2\Lambda$, $\Lambda$ being any diagonal unitary matrix.

Is there a way to calculate $\pi_1(M)$ and $\pi_2(M)$?

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