Project points onto 4D hypercone

Question

Let $p \in \mathbb R^4$ be a point and let $C$ be the $4$-dimensional unit hypercone represented with the equation $x^2 + y^2 + z^2 = w^2$. Is there an elegant way (like a closed form solution) to project a point $p$ onto $p^* \in C $, where $p^*:=\arg\min_{p_c\in C} \Vert p_c - p\Vert$? $\Vert\cdot\Vert$ is the euclidean norm.