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.