Trace-free part of the second fundamental form

Question

I found this notion on this part of a survey by S. Brendle:

What exactly is the definition (without using basis) of the “trace-free part of the second fundamental form”?

For me, the second fundamental form of a hypersurface $\Sigma$ of $S^3$ with unit normal vector field $\eta$ is, at a point $p \in \Sigma$, defined by

$$ A_p(v) = - \nabla_v \eta, \quad v \in T_p \Sigma.$$