Orthogonal Jordan idempotents

Definition

Two Jordan idempotents $e,f$ in a Jordan algebra $J$ are orthogonal when

e\circ f=0.

Their sum is then again an idempotent, because $(e+f)\circ(e+f)=e+f$ .

Euclidean interpretation

In a Euclidean Jordan algebra with trace inner product $\langle x,y\rangle=\operatorname{tr}(x\circ y)$ , two idempotents are orthogonal in the Jordan sense if and only if they are orthogonal for this inner product. For Hermitian matrices this says that the corresponding orthogonal projections have mutually orthogonal ranges.

Pairwise orthogonal primitive idempotents which sum to the unit form a Jordan frame.

References

Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994, Chapter III, §1. Publisher record.
Kevin McCrimmon, A Taste of Jordan Algebras, Springer, 2004, §13.1. Publisher record.