= Real embedding of SU(n) in SO(2n)
{title2=$A+iB\mapsto\begin{pmatrix}A&-B\\B&A\end{pmatrix}$}
Writing complex coordinates as $z=x+iy$ converts the <unitary representation> $U=A+iB$ into the indicated real <matrix>. It preserves the <Euclidean metric> because $U^\dagger U=I$, and its real <determinant> is $|\det_{\mathbb C}U|^2=1$. Thus even $U(n)$ embeds in $SO(2n)$. The complexified defining real <representation> restricts as $\mathbf n\oplus\bar{\mathbf n}$, whereas the underlying real space is the <realification of a complex representation> of dimension $n$.
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