I'm currently getting into studying optimization problems over symmetric cones (NSCP) and I'm having some trouble to understand something.

Let me first give some context, sorry if it is repetitive to you: An Euclidean Jordan algebra $\mathbb{J}$ is a vector space equipped with a Jordan product $\circ$ and an inner product $\langle\cdot ,\cdot\rangle$. Moreover, if we consider the linear operator $L_x(y) = x\circ y$ for each $x,y\in\mathbb{J}$, we have that $$\langle L_x(y),z\rangle = \langle y, L_x(z)\rangle \text{ for each } x,y,z\in\mathbb{J}.$$ That is, the operator $L_x(\cdot)$ is self-adjoint witih respect to $\langle\cdot,\cdot\rangle$.

My question is how do we relate the spectral decomposition of an element $x\in\mathbb{J}$ with the spectral decomposition of the operator $L_x(\cdot)$. I've been investigating the main examples ($\mathbb{R}^n$, $\mathbb{S}^n$, and $\mathbb{R}^n\times\mathbb{R}$) and couldn't find any general rule regarding this topic.

More generally, what is the relation between Jordan frames and basis from the underlying vector space, if there is any?

Has anyone wondered about these questions before and could figure it out? Any help, guidelines and references are welcome.

Thanks for your attention!