Eq. (40) — the partial trace over the site intertwines $V\otimes L$ conjugation with $V$ conjugation

Let $a$ and $b$ be finite index sets and consider operators on $\mathbb C^a\otimes\mathbb C^b$, where the second factor plays the role of the last NRG site. Write $\operatorname{Tr}_{\mathrm{site}}$ for the partial trace over the second factor. For every matrix $V$ on $\mathbb C^a$, every unitary matrix $L$ on $\mathbb C^b$ (that is, $L^\dagger L = 1$) and every operator $R$ on $\mathbb C^a\otimes\mathbb C^b$,

In the paper the symmetry transformation on the first $n$ sites factors as $U(g) = L(g)V(g)$, with $L(g)$ acting on site $n$ and $V(g)$ on sites $0,\dots,n-1$; this identity is the step that moves the symmetry through the trace over site $n$.

Formalization Note The statement holds for an arbitrary $V$; only $L$ needs to be unitary. Operators are indexed with the block factor first and the site factor second; the partial trace is the published WildeQIT.partialTraceRight.