# Representations with noticeable property

Let $$\rho : \mathbb{R}^n\to \mathfrak{so}(2m)$$ be a faithful representation of the commutative Lie algebra $$\mathbb{R}^n$$ into the Lie algebra of skew-symmetric matrices (a Lie algebra morphism). Since $$\rho(\mathbb{R}^n)$$ is contained in the maximal abelian subalgebra of $$\mathfrak{so}(2m)$$, then $$n\leq m$$ and there is an orthonormal basis $$d_1,\dotsc,d_{2m}$$ of $$\mathbb{R}^{2m}$$ such that for all $$i=1,\dotsc,n$$ and $$j=1,\dotsc,m$$ $$\rho(e_i)(d_{2j-1})=\lambda_{ij}\,d_{2j}\ \text{and}\ \rho(e_i)(d_{2j})=-\lambda_{ij}\,d_{2j-1}$$ for some real numbers $$\lambda_{ij}$$. In addition, we assume that for all $$i=1,\dotsc,m$$, there is $$k=1,\dotsc,n$$ such that $$\lambda_{ki}\neq0$$.

The degeneracy property of this representation is as follows $$$$\label{deg}\tag{1} \exists i This is a well defined notion (coordinate free), according to Degenerate representation.

I would like to study those representations for which $$$$\label{eigen}\tag{2} \exists \ell,i i.e. $$\lambda_{k\ell}^2$$ is an eigenvalue of the matrix: $$A=\begin{pmatrix} \left(\lambda_{ki}^2+\lambda_{kj}^2\right)&-2\lambda_{ki}\lambda_{kj}\\ -2\lambda_{ki}\lambda_{kj}&\left(\lambda_{ki}^2+\lambda_{kj}^2\right) \end{pmatrix}=\begin{pmatrix} -\lambda_{kj}&\phantom{-}\lambda_{ki}\\ \phantom{-}\lambda_{ki}&-\lambda_{kj} \end{pmatrix}^2.$$

Is this well defined (coordinate free)? In other words, is there a particular representation in this family for which $$\eqref{eigen}$$ is satisfied regardless of the chosen basis of $$\mathbb{R}^n$$?