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Definition

so*(12) — quaternionic orthogonal real form

so*(12) — quaternionic orthogonal real form.

The real Lie algebra so∗(12)\mathfrak{so}^*(12) can be realized using quaternionic matrices as

{X∈M6(H):X∗J+JX=0},J=(0I3−I30),[X,Y]=XY−YX.\{X\in M_6(\mathbb H):X^*J+JX=0\},\qquad J=\begin{pmatrix}0&I_3\\-I_3&0\end{pmatrix},\qquad [X,Y]=XY-YX.

Conjugate transpose uses quaternionic conjugation, and the scalar field of this is R\mathbb R.

Block count and naming

Writing X=(ABCD)X=\begin{pmatrix}A&B\\C&D\end{pmatrix} gives D=−A∗, B=B∗, C=C∗D=-A^*,\ B=B^*,\ C=C^*. The block AA contributes 36 real parameters; each Hermitian quaternionic 3-by-3 block contributes 15. Hence the dimension is 36+15+15=6636+15+15=66. The real part of the trace vanishes automatically. The complexification is so(12,C)\mathfrak{so}(12,\mathbb C).

References
  1. 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.