Even Dimensional Spheres and Lie Algebra Inclusions

Question

The even dimensional spheres are homogeneous spaces of the form $S^{2n} = SO(2n+1)/SO(2n)$. What is the inclusions of Lie algebras $\frak{so}(2n) \hookrightarrow \frak{so}(2n+1)$ dual to the inclusion of groups $SO(2n) \hookrightarrow SO(2n+1)$. That is to say, where does the inclusion send the basis elements $E_i,F_i,K_i \in \frak{so}(2n)$, for $i =1,2$.

Answer 1

We have $\mathfrak{so}_n(K)=\{A \in M_n(K) \mid A^T+A=0\}$, and we can embed a skew-symmetric matrix of size $2n$ into one of size $2n+1$ in the obvious way, just by adding a zero column and zero row.