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

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

\mathfrak{so}(2,2)=\{X\in M_{4}(\mathbb R):X^T\eta+\eta X=0\},\qquad \eta=\operatorname{diag}(I_{2},-I_{2}).

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 $6$ . This is a vector-space count, not an assertion that compact and indefinite brackets are isomorphic. The equal-signature form is split.

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