so(2,C) — complex orthogonal Lie algebra

The complex orthogonal Lie algebra $\mathfrak{so}(2,\mathbb C)$ consists of skew-symmetric $2\times2$ matrices, with commutator bracket:

\mathfrak{so}(2,\mathbb C)=\{X\in M_{2}(\mathbb C):X^T+X=0\},\qquad [X,Y]=XY-YX.

The transpose has no complex conjugation: the preserved form is complex bilinear, not Hermitian.

Dimension and structure

The independent entries above the diagonal give complex dimension $1$ ; a basis is $E_{ij}-E_{ji}\ (1\leq i<j\leq 2)$ . There is one generator and its self-bracket is zero, so this algebra is abelian.

The family definition explains its general construction.

Scalar convention

This record retains $\mathbb C$ as its defining field. Its underlying real vector space has twice the complex dimension; real-linear maps need not be complex-linear.