We consider linear functionals L defined on finite dimensional subspaces
V of R[x1, . . . , xn] and being square positive, L(p2) _ 0 for arbitrary
polynomials p with p2 2 V . We give a constructive proof how
to extend such an L to a square positive L_ defined on R[x1, . . . , xn]
keeping the real ideal A(L_) := {p | L_(p2) = 0} as small as possible.
We also study by examples how to obtain maximal real ideals A(L_).
As a byproduct we obtain a criterion for deciding that a given set
of polynomials has only real points as common zeros and a complete
description of all cubature formulas of degree 3 with four nodes and
positive weights for the standard integral over the triangle.