# Criterion for Miller index to specify plane and its normal at the same time?

Suppose I have an arbitrary set of Miller indices $$(hkl)$$ for a 3-d crystal lattice defined by the primitive vectors $$\{\vec{a}_i\}$$ for $$i=1,2,3$$. We know that the plane labeled by $$(hkl)$$ must always intersect the crystal axes at points $$\frac{\vec{a}_1}{h}$$, $$\frac{\vec{a}_2}{k}$$, and $$\frac{\vec{a}_3}{l}$$. It is not always the case that $$[hkl]$$ is a vector normal to the plane.

However, when the lattice is a simple cubic lattice, it happens to be that $$[hkl]$$ is indeed perpendicular to the plane. I was trying to think about under what conditions the lattice type has to satisfy for this to hold, more generally; but I couldn't exactly wrap my head around it. Would anyone have some thoughts to share?