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

For an integer $n\geq1$ , the indefinite orthogonal real Lie algebra $\mathfrak{so}(n,n)$ preserves a symmetric bilinear form with $n$ positive and $n$ negative directions infinitesimally:

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

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 $n(2n-1)$ . 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 $\mathfrak{so}(n,n)$ .