The Lie algebra of ##\frak{so}(3)## without complexification

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TL;DR Summary

Can the Lie algebra of $\frak{so}(3)$ (or $\frak{su}(2)$) be formulated without complexification utilizing the Cartan subalgebra?

All of the formulations of the Lie algebra of $\frak{so}(3)$ (or $\frak{su}(2)$) utilizing raising/lowering operators that I have seen in the literature involve complexification to $\frak{su}(2) + i \frak{su}(2) \cong \frak{sl}(2,\mathbb{C})$. I have found explicit derivations in a particular representation, but none from the Cartan subalgebra.

Can one derive a formulation of the Lie algebra of $\frak{so}(3)$ utilizing the Cartan subalgebra and root vectors without complexification? If so, where can I find it?

 

[PeterDonis]

Moderator's note: Thread moved to the linear algebra math forum.

 

[fresh_42]

$$\mathfrak{so}(3)\cong \left(\mathbb{R}^3,\times\right)=\bigl\langle U,V,W\,|\,[U,V]=W,[V,W]=U,[W,U]=V \bigr\rangle $$
We need to break that symmetry in order to get the representation via the root system that specifies the generator of the Cartan subalgebra which is not symmetric. We therefore need complex numbers for the isomorphism. You can find the basis transformations at
https://www.physicsforums.com/insights/journey-manifold-su2-part-ii/

For a description of how the root system works, see
https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-structures/
and the general basis of real orthogonal Lie algebras here:
https://www.physicsforums.com/insig...hogonal-Lie-Algebra-On-Odd-Dimensional-Spaces