Fredkin-chain conjecture on the B- and C-type symmetry algebra

Let $\mathfrak{g}$ be the Lie algebra generated by the raising and lowering operators $\Sigma^\pm$ of the periodic Fredkin chain with $N$ sites, and set $n=\lceil N/2\rceil$. B- and C-type symmetry-algebra conjecture. The rank of $\mathfrak{g}$ is $n$ and

$$\mathfrak{g}=\begin{cases} B_n=\mathfrak{so}_{2n+1}, & N=2n-1, \\ C_n=\mathfrak{sp}_{2n}, & N=2n. \end{cases}$$

This identifies the algebra generated by the proposed raising and lowering operators with a classical Lie algebra whose type is determined by the parity of the chain length. The conclusion was obtained by studying chains with $N\leq 10$ and remains conjectural.