p. 9 (with eq. (11), p. 4) — V′(n−1, n) = Φ⁺_{n−1,n}, the unnormalized maximally entangled projector on the last two systems

Let $(n\ n-1)$ be the transposition of the last two of $n$ systems of $\mathbb C^d$ and $V'(n\ n-1)$ the partial transpose, on the last system, of its permutation operator. Then

the identity on the first $n-2$ systems tensored with the unnormalized projector onto the maximally entangled vector $\sum_l|ll\rangle$ of systems $n-1,n$. Entrywise: the matrix element between basis strings $x$ and $y$ is $1$ if $x_s=y_s$ for all $s\le n-2$, $x_{n-1}=x_n$ and $y_{n-1}=y_n$, and $0$ otherwise.

This identifies the only generator of $\mathcal A$ on which the partial transposition acts non-trivially, and it is the operator through which every $v^{ab}_{ij}(\alpha)$ is built.

Formalization Note. $n=m+2$, systems are 0-based, so systems $n-1,n$ are the indices $m$ and $m+1$ and the first $n-2$ systems are the images of Fin.castLE from Fin m. On p. 9 $\Phi^+$ denotes the unnormalized projector (eq. (11) calls $|\widetilde\Phi^+\rangle$ "unnormalized"); the statement is the unnormalized form, which is $d\cdot\mathbb 1\otimes|\Phi^+\rangle\langle\Phi^+|$ for the normalized $|\Phi^+\rangle$ of Remark 3.