# An Explicit isomorphism between the orthogonal Lie algebras $\mathfrak{so}_n$ and the Lie algebras of type $B_n$ or $D_n$.

I know from many sources, including Samelson's Notes on Lie Algebras, that the Lie algebras of even size skew-symmetric matrices, $$\mathfrak{so}_{2n}$$, are isomorphic to the Lie algebras of type $$D_n$$ and the odd size ones are isomorphic to the Lie algebras of type $$B_n$$. I cannot find an explicit isomorphism at the basis level anywhere. To clarify, in the case of $$\mathfrak{so}_{2n}$$ for example, I'm hoping for an explicit isomorphism that sends the basis elements $$\{e_{i,j}-e_{j,i}\mid 1\leq i for $$\mathfrak{so}_{2n}$$ to a $$2n\times 2n$$ matrix of the form $$\left(\begin{matrix}A&B\\C&-A^T\end{matrix}\right)$$ where $$A,B,C\in\mathfrak{gl}_n$$, $$B=-B^T$$, and $$C=-C^T$$, which is the form of the elements of $$D_n$$ given in Humphreys Introduction to Lie Algebras and Representation Theory. I would also love one for the odd size matrices to the basis for $$B_n$$ given in the same source. Could someone please provide these isomorphisms?

$$S_1=\pmatrix{1 & 0 &\dots & 0\\ 0&1& &0\\ && \ddots &\\ 0&0&&1}$$ (giving the skew-symmetric matrices) to the one with, depending on the parity of $$n$$, $$S_2=\begin{pmatrix} 1&0&0\\ 0&O&I_l\\ 0&I_l&O \end{pmatrix} \hspace{.5in}\text{or}\hspace{.5in} S_2=\begin{pmatrix} O&I_l\\ I_l&O \end{pmatrix}$$
(with $$l=\lfloor n/2\rfloor$$) or some scalar multiple thereof, giving the split forms as described by Humphreys, where the diagonal matrices form a nice CSA and the roots are visible.
Note that in the other answer, $$S_2$$ was a bit different again, but that $$S_2$$ and Humphreys' $$S_2$$ are easily seen to have an easy base change between them which works over almost any field. Meaning that like in the other answer, this entire base change should work, i.e. you can find a matrix $$P$$ with $$^tPS_1P=S_2$$, if and only if the field over which we are working contains a square root of $$-1$$, which will come up in the base change matrix and in the explicit isomorphism. In particular, the described Lie algebras are indeed isomorphic e.g. over $$\mathbb C$$, but not over $$\mathbb R$$. (Cf. Two Definitions of the Special Orthogonal Lie Algebra, note my comment to the accepted answer.)