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We show that there can be no finite list of conditional independencerelations which can be used to deduce all conditional independence implicationsamong Gaussian random variables. To do this, we construct, for each $n> 3$ afamily of $n$ conditional independence statements on $n$ random variables whichtogether imply that $X_1 \ind X_2$, and such that no subset have this sameimplication. The proof relies on binomial primary decomposition.

Gaussian conditional independence relations have no finite complete characterization