  Our purpose is to characterize the so-called horizontal Fock-Carleson type
measures for derivatives of order $k$ (we write it $k$-hFC for short) for the
Fock space as well as the Toeplitz operators generated by sesquilinear forms
given by them. The boundedness conditions for such operators are found. We
introduce real coderivatives of $k$-hFC type measures and show that the
C*-algebra generated by Toeplitz operators with the corresponding class of
symbols is commutative and isometrically isomorphic to certain $C^*$-subalgebra
of $L_{\infty}(\mathbb{R}^{n})$. The above results are extended to measures
that are invariant under translations along Lagrangian planes.
