Vector Notation of $nablacdotleft(left(vec{v}cdotnablaright)vec{v}right)$ where $0=nablacdotvec{v}$

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
$$nablacdotleft(left(vec{v}cdotnablaright)vec{v}right)=left(vec{v}cdotnablaright)left(nablacdotvec{v}right)+sum_{i=1}^{3}sum_{j=1}^{3}left(frac{partial v_j}{partial x_i}right)left(frac{partial v_i}{partial x_j}right)=sum_{i=1}^{3}sum_{j=1}^{3}left(frac{partial v_j}{partial x_i}right)left(frac{partial v_i}{partial x_j}right)=Trace{left(left(nablavec{v}right)^2right)}$$
To clarify,
$$left(vec{A}cdotnablaright)vec{B}=A_xleft(frac{partialvec{B}}{partial x}right)+A_yleft(frac{partialvec{B}}{partial y}right)+A_zleft(frac{partialvec{B}}{partial z}right)$$