I stumbled across exercises asking to prove the following isomorphisms:
- $\mathfrak{sl}_2(\mathbb{R}) \cong \mathfrak{so}_{2,1}(\mathbb{R})$
- $\mathfrak{sl}_2(\mathbb{C}) \cong \mathfrak{so}_{3,1}(\mathbb{R})$
- $\mathfrak{so}_{2,2}(\mathbb{R}) \cong \mathfrak{sl}_2(\mathbb{R}) \oplus \mathfrak{sl}_2(\mathbb{R})$
I actually tried finding the isomorphisms. For the first one, I chose the basis $$i = \begin{pmatrix} 0&1&0\\-1&0&0\\0&0&0 \end{pmatrix}, \, j = \begin{pmatrix} 0 & 0 & 1\\ 0 & 0 & 0\\ 1 & 0 & 0 \end{pmatrix}, \, k=\begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0 \end{pmatrix}$$ and computed $[i,j] = -k$, $[i,k] = j$ and $[j,k]=i$. Of course the aim is to relate this to the commutators of the usual $\mathfrak{sl}_2$-triple. By adding $i$ and $j$, I get $2[i+j,k] = i+j$ which already looks promising, but I struggle to go on.
I don't really see the point in doing these calculations and I don't think spending hours trying to figure out some nice way to add the basis elements such that all works out is a very valuable exercise.
Could anyone give the isomorphisms explicitly or at least know a reference of it?
Let's start with the first example and use your $i,j,k$ for the $SO(2,1,R)$ matrices. The analogous $SL(2,R)$ matrices with the same commutators are multiples of the Pauli matrices $$ j = \frac 12 \pmatrix{ 0&1\\1&0 },\,\, i = \frac 12 \pmatrix{ 0&-1\\1&0 },\,\, k = \frac 12 \pmatrix{ 1&0\\0&-1 }, $$ You may easily verify that these three matrices anticommute with each other and the same commutators hold $$ [i,j]=-k,\,\,[i,k]=j,\,\,[j,k]=i $$ which is the only nontrivial part of the isomorphism.
The second isomorphism may be obtained simply by realizing that $SL(2,C)$ is the complexification of $SL(2,R)$ – we allow the coefficients to be complex, not real. And similarly $SO(3,1)$ is isomorphic to the complexification of $SO(3,R)$ or $SO(2,1,R)$, too. The latter statement is easily seen if we write the generators of $SO(3,1,R)$ as $J_{ij}$ which are $ij$-antisymmetric and write $$ K^\pm_3 =\frac 12 ( J_{12} \pm J_{34}) $$ and the 123-cyclic permutations of this expression. With this definition, $K^\pm_{1,2,3}$ form two separate algebras that are isomorphic to $SO(3)$ or $SO(2,1)$, one only has to distinguish the signs and reality conditions.
For $SO(3,1)$, the generators $K^+_i$ are the Hermitian conjugates to $K^-_i$, so the coefficients have to be complex conjugate to each other as well, and we get one complexified $SL(2,C)$ algebra. For $SO(4,R)$ or $SO(2,2,R)$, which is your last cases, one finds out that $K^+$ and $K^-$ are independent of each other, even through Hermitian conjugation, and finds out that these two algebras are isomorphic to $SU(2) \times SU(2)$ or $SO(2,1,R)\times SO(2,1,R)$, respectively (these two cases differ by some signs already apparent in the 3-dimensional algebras).
It's useful to go through the calculation using a particular form of the matrices at least once. But the final result may be expressed in many ways and is independent of the choice of the bases etc. And it may be seen to hold in many ways. There are really not too many 3- or 6-dimensional Lie algebras.
Those results are basics of Lie group/algebra theory and their importance especially in physics cannot be overstated.