A generalization of Cartan's theorem on isoparametric cubics

We generalize the well-known result of É. Cartan on isoparametric cubics by showing that a homogeneous cubic polynomial solution of the eiconal equation $\vert\nabla f\vert^2=9\vert x\vert^4$ must be rotationally equivalent to either $x_n^3-3x_n(x_1^2+\ldots+x_{n-1}^2)$ or to one of four exceptional Cartan cubic polynomials in dimensions $n=5,8,14,26$.