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A set $S\subset \mathbb{R}^n$ is a nonnegativity witness for a set $U$ ofreal homogeneous polynomials if $F$ in $U$ is nonnegative on $\mathbb{R}^n$ ifand only if it is nonnegative at all points of $S$. We prove that the union ofthe hyperplanes perpendicular to the elements of a root system $\Phi\subseteq\mathbb{R}^n$ is a witness set for nonnegativity of forms of low degree whichare invariant under the reflection group defined by $\Phi$. We prove that ourbound for the degree is sharp for all reflection groups which containmultiplication by $-1$. We then characterize subspaces of forms of arbitrarilyhigh degree where this union of hyperplanes is a nonnegativity witness set.Finally we propose a conjectural generalization of Timofte's half-degreeprinciple for finite reflection groups.

Test Sets for Nonnegativity of Polynomials Invariant under a Finite Reflection Group