Vector Notation of $\nabla\cdot\left(\left(\vec{v}\cdot\nabla\right)\vec{v}\right)$ where $0=\nabla\cdot\vec{v}$

Question

I want a simplified way of writing the above expression in a clear way using vector notation. It is given that $\vec{v}$ has a divergence of 0. I’ve found that $$\nabla\cdot\left(\left(\vec{v}\cdot\nabla\right)\vec{v}\right)=\left(\vec{v}\cdot\nabla\right)\left(\nabla\cdot\vec{v}\right)+\sum_{i=1}^{3}\sum_{j=1}^{3}\left(\frac{\partial v_j}{\partial x_i}\right)\left(\frac{\partial v_i}{\partial x_j}\right)=\sum_{i=1}^{3}\sum_{j=1}^{3}\left(\frac{\partial v_j}{\partial x_i}\right)\left(\frac{\partial v_i}{\partial x_j}\right)=Trace{\left(\left(\nabla\vec{v}\right)^2\right)}$$ To clarify, $$\left(\vec{A}\cdot\nabla\right)\vec{B}=A_x\left(\frac{\partial\vec{B}}{\partial x}\right)+A_y\left(\frac{\partial\vec{B}}{\partial y}\right)+A_z\left(\frac{\partial\vec{B}}{\partial z}\right)$$