Singular solutions from simple Euclidean Jordan algebras

For every finite-dimensional simple Euclidean Jordan algebra $V$ of rank at least three, we realize the function $\operatorname{tr}(x^3)/|x|$ on its traceless subspace $V_0$ as a $C^{1,1}\setminus C^2$ viscosity solution of a uniformly elliptic equation $F(D^2 u)=0$ in $\mathbb{R}^{\operatorname{dim}V_0}$. This unifies and extends earlier constructions from Cartan isoparametric cubics, Hsiang minimal cubics, and traceless real symmetric matrices.