su(3,3) — indefinite special unitary Lie algebra

The indefinite special unitary Lie algebra $\mathfrak{su}(3,3)$ is the real Lie algebra

\mathfrak{su}(3,3)=\{X\in M_6(\mathbb C):X^*I_{3,3}+I_{3,3}X=0,\ \operatorname{tr}X=0\},\qquad I_{3,3}=\operatorname{diag}(I_3,-I_3).

Its bracket is $[X,Y]=XY-YX$ and scalar multiplication is over $\mathbb R$ .

Dimension

Multiplication by $I_{3,3}$ identifies the skew-adjoint equation with the usual skew-Hermitian equation. Its trace is purely imaginary, so trace zero imposes one real condition on 36 parameters. The real dimension is therefore 35.

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.