I want to write down this thing in a matrix-ish way:
$$ A_{ij} = \alpha\ v\nabla_i\nabla_ju + \beta\ (\nabla_i v)(\nabla_j u) + \gamma\ (\nabla_j v) (\nabla_i u) + \delta\ u\nabla_i\nabla_jv $$
The closest form I come up with: $$ A_{ij} = \pmatrix{v\nabla_i\\\nabla_iv}^T \pmatrix{\gamma& \delta\\\alpha& \beta} \pmatrix{u\nabla_j\\\nabla_ju} = \\ \underbrace{\alpha\ v\nabla_i \nabla_j u + \beta\ (\nabla_iv)(\nabla_ju)}_{\text{nice}} + \underbrace{\gamma\ v(\nabla_iu)\nabla_j + \delta\ (\nabla_iv)u\nabla_j}_{\text{nonsense}} $$
Could you suggest a notation that reflect the similarity to bilinear form?