so(p,q) — indefinite orthogonal real Lie algebra

For nonnegative integers $p,q$ with $p+q\geq1$ , the indefinite orthogonal real Lie algebra $\mathfrak{so}(p,q)$ is defined by

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

Its bracket is $[X,Y]=XY-YX$ . The transpose is real transpose.

Signature, dimension and scalars

The signature convention is $p$ positive and $q$ negative directions. Negating the entire form leaves the matrix Lie algebra unchanged, and exchanging the two coordinate blocks identifies signatures $(p,q)$ and $(q,p)$ . The matrix $\eta X$ is skew-symmetric, so counting entries above its diagonal gives the real dimension.

The scalar field of this Lie algebra is $\mathbb R$ .

References

Barton and Sudbery, Magic squares and matrix models of Lie algebras, §2, pp. 5–8, equations (2.19)–(2.27); quaternionic and indefinite matrix conventions.