so(2,R) — real orthogonal Lie algebra

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

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

The form preserved here is the positive-definite real form $x_1^2+\cdots+x_2^2$ .

Dimension and structure

The independent entries above the diagonal give real 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 $\mathbb R$ as its defining field. Its complexification is a different catalogue object; a matrix presentation over complex numbers does not change this field automatically.