Fusion rules from the Norton inequality

We study fusion rules forced by the Norton inequality in commutative non-associative real algebras equipped with a Frobenius form. We answer a question of T. M. Mudziiri Shumba and S. Shpectorov concerning whether the eigenspace $A_0(e)$ associated with an arbitrary idempotent $e\in A$ must be a subalgebra of $A$. If the Frobenius form is non-degenerate, as in Majorana algebras, then for every idempotent $e\in A$, both $A_0(e)$ and $A_1(e)$ are subalgebras of $A$, and $$A_0(e)A_1(e)\subseteq A_{1\mathrm{/}2}(e)\mathrm{.}$$ In the degenerate case, the corresponding inclusions hold modulo the radical of the Frobenius form. We also give an explicit axial algebra with a degenerate Frobenius form satisfying the Norton inequality for which $A_0(e)$ is not a subalgebra for one of its axes.