  By using a nonassociative algebra argument, we prove that $u\equiv0$ is the
only cubic homogeneous polynomial solution to the $p$-Laplace equation
$\mathrm{div} |Du|^{p-2}Du(x)=0 $ in $\mathbb{R}^n$ for any $n\ge2$ and
$p\not\in\{0,2\}$.
