Any 2x2 PSD biquadratic polynomial is a sum of three squares of quadratic polynomials, via a variable-reduction technique and constructive decompositions.
Choi, ``Positive semidefinite biquadratic forms'', Linear Algebra and Its Applications 12 (1975) 95-100
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Postive Semidefinite and Sum of Squares Biquadratic Polynomials
Any 2x2 PSD biquadratic polynomial is a sum of three squares of quadratic polynomials, via a variable-reduction technique and constructive decompositions.