  The gaussoid axioms are conditional independence inference rules which
characterize regular Gaussian CI structures over a three-element ground set. It
is known that no finite set of inference rules completely describes regular
Gaussian CI as the ground set grows. In this article we show that the gaussoid
axioms logically imply every inference rule of at most two antecedents which is
valid for regular Gaussians over any ground set. The proof is accomplished by
exhibiting for each inclusion-minimal gaussoid extension of at most two CI
statements a regular Gaussian realization. Moreover we prove that all those
gaussoids have rational positive-definite realizations inside every
$\varepsilon$-ball around the identity matrix. For the proof we introduce the
concept of algebraic Gaussians over arbitrary fields and of positive Gaussians
over ordered fields and obtain the same two-antecedental completeness of the
gaussoid axioms for algebraic and positive Gaussians over all fields of
characteristic zero as a byproduct.
