I'm having trouble understanding a few comments is Kühnel's "Differential Geometry: curves - surfaces - manifolds", in page $118$, about the definition of the second-fundamental form in Lorentz-Minkowski space $\Bbb L^3$:
... the second fundamental form is just the normal component of the matrix of the second derivatives (cf. $3.10$), and is therefore vector-valued. In the Euclidean case, this was irrelevant, we simply set there ${\rm II}(X,Y) = {\rm I}(LX,Y)$ and viewed the second fundamental form as the scalar factor of this compared to $\nu$. ...
From what little research I've made and from what follows after this, I suppose that he meant $${\rm II}(X,Y):= -\langle {\rm d}\nu(X),Y \rangle \, \nu,$$ but my problem is: I understand that you can form a matrix taking the normal components, that is, ${\rm II} = (h_{ij}) = (-\langle {\rm d}\nu({\bf x}_i),{\bf x}_j\rangle)$, but what would it mean, in his words, to take the normal component of a matrix?
(here $\nu$ is an unit normal, and $L = -{\rm d}\nu$ is the Weingarten map)