The timelike cone is the open set of nonzero tangent vectors with timelike squared norm. It has future and past components once a time orientation is chosen. Homogeneity extends an identity for unit timelike vectors to this whole open set. In particular, if a symmetric quadratic form vanishes on every timelike vector, its polynomial vanishes on an open set and therefore vanishes identically. This observation proves the converse from geodesic first integrals to the Killing equation.
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