Remark. The radical blocks an unqualified extension theorem [fcap-002C]

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For \(Q(a,b,c)=a^2-b^2\) on \(\mathbb R^3\), with \(e_1,e_2,e_3\) the standard coordinate basis, the radical is \(\mathbb R e_3\). The zero-form lines \(\mathbb R e_3\) and \(\mathbb R(e_1+e_2)\) are isometric by \(e_3\mapsto e_1+e_2\), but this map has no ambient isometric extension. Every ambient isometry preserves the radical, whereas \[B_Q(e_1+e_2,e_1)=2,\] so \(e_1+e_2\) is not radical. This example lies outside both guards above: the ambient form is degenerate, and the source line is not regular.

For the projective-geometric-algebra form \(Q_{\mathrm {PGA}}(a,b,c)=a^2+b^2\), the positive, negative, and radical dimensions are \((2,0,1)\), with radical \(\mathbb R e_3\). Its radical quotient is a two-dimensional positive-definite space. Quotienting preserves the positive and negative indices QuadraticForm.sigPos_lift_radical QuadraticForm.sigNeg_lift_radical, but it changes the quadratic space. It therefore cannot silently replace the original degenerate Clifford algebra or its motor carrier. Failure of the classical ambient nondegeneracy hypothesis does not show that extension fails for PGA; TauCeti's regular-subspace theorem may still apply there.