We provide a construction of an embedded minimal hypersurface in $\mathbb{R}^4$ with $D-4$ symmetry group and discuss its inner structure. The hypersurface has a discrete set of singularities of the Clifford cone type. We are also going to discuss some possible generalizations of this example

A classification and construction of algebraic minimal cones of degree higher or equal than 3 remains a long-standing difficult problem. We will discuss the first non-trivial case of cubic minimal cones, and, radial eigencubics in particular (the study initiated earlier by Hsiang in [Hsia67]). We show that there are hidden Clifford and Jordan algebra structures associated with the radial eigencubics, and specify the latter in the context of the eiconal type equation $\|Du(x)\|^2=9\|x\|^4$ which plays a crucial role in the classification [Tk10]. In particular, we establish a natural bijective correspondence between cubic solutions of the eiconal type equation with a general (not necessarily Euclidean) norm $\|\cdot\|$ and semi-simple rang three Jordan algebras. We also illustrate the appearance of cubic Jordan algebras by some related problems discussed recently in [NTV12].
[Hsia67] Wu-yi Hsiang, Remarks on closed minimal submanifolds in the standard Riemannian m-sphere, J. Differential Geometry 1 (1967), 257–267.
[NTV12] N. Nadirashvili, V.G. Tkachev, and S. Vladut, A non-classical solution to a Hessian equation from Cartan isoparametric cubic, Adv. Math. 231 (2012), no. 3-4, 1589–1597.
[Tk10] V. G. Tkachev, A generalization of Cartan's theorem on isoparametric cubics, Proc. Amer. Math. Soc. 138 (2010), no. 8, 2889–2895.