sl(2,C) with scalars restricted to R

The underlying real Lie algebra $\mathfrak{sl}(2,\mathbb C)_{\mathbb R}$ has the same trace-zero complex matrices and commutator bracket as $\mathfrak{sl}(2,\mathbb C)$ , but only real scalar multiplication is part of its structure.

Retained and forgotten structure

Its real dimension is 6. Multiplication by $i$ is an additional real-linear operator $J$ with $J^2=-1$ and $[JX,Y]=J[X,Y]=[X,JY]$ . A real-linear homomorphism is complex-linear precisely when it commutes with $J$ .

This operation differs from taking a real form: $\mathfrak{sl}(2,\mathbb R)$ has only 3 real dimensions.