so(3,3) — indefinite orthogonal real Lie algebra

The indefinite orthogonal real Lie algebra $\mathfrak{so}(3,3)$ preserves a symmetric bilinear form with $3$ positive and $3$ negative directions infinitesimally:

\mathfrak{so}(3,3)=\{X\in M_{6}(\mathbb R):X^T\eta+\eta X=0\},\qquad \eta=\operatorname{diag}(I_{3},-I_{3}).

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 $15$ . 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 $\mathfrak{so}(3,3)$ .