Makt wrote:

As far as I heard (I am not sure about the precise statement) there is a classification of simple Jordan algebras of Euclidean type.

Yes, in this paper

- Pascual Jordan, John von Neumann and Eugene Wigner, On an algebraic generalization of the quantum mechanical formalism,
*Ann. Math.* **35** (1934), 29–64.

the authors proved:

**Theorem.** Every finite-dimensional Euclidean Jordan
algebra is isomorphic to a direct sum of simple ones, and these are all the simple ones:

$\mathfrak{h}_n(\mathbb{R})$: $n \times n$ self-adjoint real
matrices with $a \circ b = \frac{1}{2}(ab + ba)$.

$\mathfrak{h}_n(\mathbb{C})$: $n \times n$ self-adjoint complex
matrices with $a \circ b = \frac{1}{2}(ab + ba)$.

$\mathfrak{h}_n(\mathbb{H})$: $n \times n$ self-adjoint quaternionic matrices with $a \circ b = \frac{1}{2}(ab + ba)$.

$\mathfrak{h}_n(\mathbb{O})$: $n \times n$ self-adjoint octonionic matrices with $a \circ b = \frac{1}{2}(ab + ba)$ where $n \le 3$.

The **spin factors** $\mathbb{R}^n \oplus \mathbb{R}$, with
$$ (x,t) \circ(x', t') =
(t x' + t' x, x \cdot x' + tt'). $$

Every Euclidean Jordan algebra comes automatically with a cone of positive definite elements, a determinant function, a trace function, and much more. A good place to learn about these is here:

- Jacques Faraut and Adam Korányi,
*Analysis on Symmetric Cones*, Oxford University Press, Oxford, 1994.

The determinant on $\mathfrak{h}_3(\mathbb{O})$ is given by

$$
\det \left( \begin{array}{ccc}
\alpha & z & y^* \\
z^* & \beta & x \\
y & x^* & \gamma \end{array} \right) =
\alpha \beta \gamma - (\alpha \|x\|^2 + \beta \|y\|^2 + \gamma \|z\|^2)
+ 2 \mathrm{Re}(xyz)
$$

where $\alpha, \beta, \gamma \in \mathbb{R}$ and $x,y,z \in \mathbb{O}$. You can check that

$$ \mathrm{Re}((xy)z) = \mathrm{Re}(x(yz)) $$

for any octonions $x,y,z$, so this justifies us in writing either one as $\mathrm{Re}(xyz)$.

For more, including more references, try

- John Baez, The octonions, Section 3.4: $\mathbb{O}\mathrm{P}^2$ and the exceptional Jordan algebra,
*Bull. Amer. Math. Soc.* **39** (2002), 145–205.