Herm_3(H) Jordan algebra

The Herm_3(H) Jordan algebra is the $\mathbb R$ -vector space

\operatorname{Herm}_{3}(\mathbb H)=\{X\in M_{3}(\mathbb H):X^*=X\}, \qquad X\circ Y=\tfrac12(XY+YX),

with unit $I_{3}$ . Here $X^*=\overline X^T$ , using the standard conjugation of the coefficient algebra.

Coordinates and dimension

The diagonal contributes $3$ scalar coordinates, and the off-diagonal pairs contribute $12$ , giving dimension $15$ over $\mathbb R$ .

Jordan identity and range

The fixed subspace is closed under $X\circ Y$ , because $(XY)^*=Y^*X^*$ . Associativity of the ambient matrix algebra proves the Jordan identity, so the construction works in every positive size.