The unbroken spectrum of Frobenius seaweeds II: type-B and type-C

Analogous to the Type-$A_{n-1}=\mathfrak{s}\mathfrak{l}(n)$ case, we show that if $\mathfrak{g}$ is a Frobenius seaweed subalgebra of $B_n=\mathfrak{s}\mathfrak{o}(2n+1)$ or $C_n=\mathfrak{s}\mathfrak{p}(2n)$, then the spectrum of the adjoint of a principal element consists of an unbroken set of integers whose multiplicities have a symmetric distribution.