Lemma: Complex bilinear response extension

For real smooth $a,b,c,d$, define

This is the unique complex bilinear extension of $B$. It satisfies

and preserves zero initial data.

Proof. Complex bilinearity forces the four-term expansion (eq:BC). Conversely, real bilinearity of $B$ verifies additivity and multiplication by $\mathrm{i}$ in each slot, so the formula defines a complex bilinear map.

The operator $L$ has real coefficients. Apply $L$ to the four terms and use Lemma lem:realB. The result is the same four-term extension of the polarized source. Initial data are preserved term by term.