## Decomposition of the sum of three vectors

Determine parameters $$a, b \in \mathbb K$$ so that for each quadratic form $$g$$ at $$V$$ above $$\mathbb K$$ and any three vectors $$\mathbf u, \mathbf v, \mathbf w \in V$$ holds: $$g(\mathbf u+\mathbf v+\mathbf w)=ag(\mathbf u+\mathbf v)+ag(\mathbf u+\mathbf w)+ag(\mathbf v+\mathbf w)+bg(\mathbf u)+bg(\mathbf v)+bg(\mathbf w)$$
We expend both sides using a bilinear form $$f$$ and compare the coefficients. $\begin{eqnarray*} && g(\mathbf u+\mathbf v+\mathbf w) \\ &=& f(\mathbf u+\mathbf v+\mathbf w,\mathbf u+\mathbf v+\mathbf w)\\ &=& f(\mathbf u,\mathbf u)+ f(\mathbf u,\mathbf v)+ f(\mathbf u,\mathbf w)+ f(\mathbf v,\mathbf u)+ f(\mathbf v,\mathbf v)+ f(\mathbf v,\mathbf v)+ f(\mathbf w,\mathbf u)+ f(\mathbf w,\mathbf v)+ f(\mathbf w,\mathbf w)\\ \end{eqnarray*}$ $\begin{eqnarray*} && a(g(\mathbf u+\mathbf v)+g(\mathbf u+\mathbf w)+g(\mathbf v+\mathbf w))+b(g(\mathbf u)+g(\mathbf v)+g(\mathbf w))\\ &=& a(f(\mathbf u+\mathbf v,\mathbf u+\mathbf v)+f(\mathbf u+\mathbf w,\mathbf u+\mathbf w)+f(\mathbf v+\mathbf w,\mathbf v+\mathbf w))+b(f(\mathbf u,\mathbf u)+f(\mathbf v,\mathbf v)+f(\mathbf w,\mathbf w)) \\ &=& (2a+b)(f(\mathbf u,\mathbf u)+ f(\mathbf v,\mathbf v)+ f(\mathbf w,\mathbf w))\\ &&+a( f(\mathbf u,\mathbf v)+ f(\mathbf u,\mathbf w)+ f(\mathbf v,\mathbf u)+ f(\mathbf v,\mathbf v)+ f(\mathbf v,\mathbf v)+ f(\mathbf w,\mathbf u)+ f(\mathbf w,\mathbf v)) \\ \end{eqnarray*}$ Therefore $$2a+b=1$$ and $$a=1$$.
The coefficients are $$a=1$$ and $$b=-1$$.