# establishing an isomorphism

I request help with this is a question from Introduction to Lie algebra by Erdmann and Wildon.

The question asks to show that show that $so(4,\mathbf{C})\cong sl(2,\mathbf{C}) \oplus sl(2,\mathbf{C})$ by first showing that the set of diagonal matrices in $so(4,\mathbf{C})$ forms a Cartan subalgebra of $so(4,\mathbf{C})$ and determining the corresponding root space decomposition.

I have done the first part but Im having difficulty in finding the root space decomposition and using that to establish the isomorphism.

The book defines $so(4,\mathbf{C})$ to be a subalgebra of $gl(n,\mathbf{C})$ given by $$x\in gl(n,\mathbf{C}) :x^tS=-Sx$$ with $S$ taken to be the matrix with $l \times l$ blocks.

PS: this is not homework.

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Are you sure about the wording of the problem? This lie algebra consists of skew-symmetric matrices and the only such diagonal matrix is the zero matrix, so these wouldn't form a Cartan subalgebra. – Santiago Canez Dec 11 '12 at 17:53
@SantiagoCanez See the addition made. Maybe, it'll help. – sawyer Dec 11 '12 at 18:19

I will assume you mean $S = \begin{pmatrix} 0&I_2\\I_2&0 \end{pmatrix}$. If so, please edit your question accordingly.
You should have found that the Cartan subalgebra $\mathfrak h$ described is $2$-dimensional. To work out the root space decomposition, you must determine the eigenvalues and eigenvectors for the adjoint action of $\mathfrak h$ on $so(4)$, and to simplify matters you only need to see what happens for the $2$ basis elements of $\mathfrak h$. You should be able to work this out explicitly: take $E_1$ to be one basis element and write out $[E_1,X]$ for $X$ an arbitrary element of $so(4)$. (By the way, you should also be able to fully determine what an element of $so(4)$ looks like.) Then ask yourself when $[E_1,X]$ could possibly be a multiple of $X$ to find the eigenvalues and eigenvectors. You would do the same for the other basis element of $\mathfrak h$.
In the end you should find $4$ roots with a $1$-dimensional eigenspace for each, and so your root decomposition will look like $so(4) = \mathfrak h \oplus \mathfrak g_1 \oplus \mathfrak g_2 \oplus \mathfrak g_3 \oplus \mathfrak g_4$ where each $\mathfrak g_\alpha$ is $1$-dimensional. Finally you must figure out how to group these factors in the direct sum together to produce two copies of $sl(2)$ in order to produce the required isomorphism.