= Positive-diagonal QR decomposition preserves a complex structure
Let $K$ be block diagonal with $2\times2$ blocks $J=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$. If a real invertible <matrix> $A$ commutes with $K$, its <QR decomposition> $A=BC$, with $B$ orthogonal and $C$ upper triangular with positive diagonal, has both $BK=KB$ and $CK=KC$. Indeed $CKC^{-1}=B^{-1}KB$, so $E=K^{-1}CKC^{-1}$ is orthogonal and upper triangular in $2\times2$ blocks. Orthogonality forces its off-diagonal blocks to vanish. Its diagonal blocks are $J^{-1}C_iJC_i^{-1}$. For $C_i=\begin{pmatrix}a&b\\0&d\end{pmatrix}$ with $a,d>0$, this block is $\begin{pmatrix}d/a&-b/a\\-b/a&(a^2+b^2)/(ad)\end{pmatrix}$. Orthogonality of its columns forces $b=0,d=a$, and hence each block is the identity. Thus $E=I$, proving the claim.
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