In this paper, Jordan–Kronecker invariants are calculated for all nilpotent $6$- and $7$-dimensional Lie algebras. We consider the Poisson bracket family, depending on the lambda parameter on a Lie coalgebra, i.e., on the linear space dual to a Lie algebra. For some space $\mathfrak{g}$ proposed in the paper, two skew-symmetric matrices are defined for all points $x$ on this linear space. To understand the behaviour of the matrix pencil $(A - \lambda B)(x)$, we consider Jordan–Kronecker invariants for this pencil and how they change with $x$ (the latter is done for $6$-dimensional Lie algebras).