2
$\begingroup$

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.

  • 0
    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.2012-12-11
  • 0
    @SantiagoCanez See the addition made. Maybe, it'll help.2012-12-11

1 Answers 1