Picture below is from the 255th page of Huisken, Gerhard, Flow by mean curvature of convex surfaces into spheres, J. Differ. Geom. 20, 237-266 (1984). ZBL0556.53001.
In my view,
$$
A*\nabla A = g^{ia}h_{ij}\nabla _a h_{bc} \tag{1}
$$
and
$$
g^{il}\nabla_j(Hh_{kl})= g^{il}h_{kl}\nabla _j H + g^{il}H\nabla _j h_{kl} \\
=g^{il}g^{ab}h_{kl}\nabla _j h_{ab} + g^{il}g^{ab}h_{ab}\nabla _{j}h_{kl} \tag{2}
$$
I think the (2) has different from with (1). So, how to understand the step of red line in picture ?
