Euclidean Jordan algebra

Remarks

Example: the real symmetric $n \times n$ matrices with $X \circ Y = \frac{1}{2}(XY + YX)$ and $\ip{X}{Y} = \tr(XY)$. The unit is the identity matrix, and $\tr(X^2) = \sum_{i, j} X_{ij}^2 > 0$ for $X \ne 0$. Clause 3 follows from $\tr(ABC) = \tr(BCA)$: both $\tr((X \circ Y) Z)$ and $\tr(Y (X \circ Z))$ equal $\frac{1}{2}\tr(XYZ + YXZ)$.

Non-example: $\R[\varepsilon]/(\varepsilon^2)$ is a unital Jordan algebra, but no inner product satisfies clause 3. Indeed clause 3 would give $\ip{\varepsilon}{\varepsilon} = \ip{\varepsilon e}{\varepsilon} = \ip{e}{\varepsilon^2} = 0$, so the inner product would not be positive definite.

A finite-dimensional real Jordan algebra admits such an inner product exactly when it is formally real and unital; formally real finite-dimensional Jordan algebras are automatically unital.