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

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

\mathfrak{so}(1,1)=\{X\in M_{2}(\mathbb R):X^T\eta+\eta X=0\},\qquad \eta=\operatorname{diag}(I_{1},-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 $1$ . This is a vector-space count, not an assertion that compact and indefinite brackets are isomorphic. The equal-signature form is split. For signature $(1,1)$ it is one-dimensional and abelian.

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