= Solution
The change of variables in <3/b> diagonalizes the complexified action into $z\mapsto Uz$ and $\bar z\mapsto\bar U\bar z$. Consequently the conventional complex <branching rule> is
$$
\boxed{\mathbf6_{SO(6)}\downarrow SU(3)=\mathbf3\oplus\bar{\mathbf3}.}
$$
Precisely, this is a statement about the <complexification of a real vector space> carrying the real defining <representation>. Over the <real numbers>, the six-dimensional space is the <realification of a complex representation> $\mathbf3$; the conjugate summands above are not separate three-dimensional real <invariant subspaces>.
The block embedding $SO(6)\hookrightarrow SO(7)$ fixes the seventh coordinate. It adds a <trivial representation>, giving
$$
\boxed{\mathbf7_{SO(7)}\downarrow SU(3)=\mathbf3\oplus\bar{\mathbf3}\oplus\mathbf1.}
$$
The dimensions are complex dimensions after <complexification of a real vector space>, or equivalently $6+1$ as real dimensions before it.
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