so(2,1) — indefinite orthogonal real Lie algebra

The indefinite orthogonal real Lie algebra $\mathfrak{so}(2,1)$ preserves a symmetric bilinear form with $2$ positive and $1$ negative directions infinitesimally:

\mathfrak{so}(2,1)=\{X\in M_{3}(\mathbb R):X^T\eta+\eta X=0\},\qquad \eta=\operatorname{diag}(I_{2},-I_{1}).

The bracket is the matrix commutator.

Dimension and real form

Multiplication by $\eta$ identifies the defining vector space with skew-symmetric matrices, so its real dimension is $3$ . This is a vector-space count, not an assertion that compact and indefinite brackets are isomorphic. The adjoint action identifies it with $\mathfrak{sl}(2,\mathbb R)$ .

Exchanging the positive and negative coordinates gives an isomorphic Lie algebra $\mathfrak{so}(1,2)$ .