Summary: Computing the relations for the coefficients satisfied by the characteristic polynomial of the Kronecker product of a general \(n \times n\) matrix by a general \(m \times m\) matrix leads to an elimination problem that is already difficult for small values of \(n\) and \(m\). In this article we focus on the problems for \((n, m) \in \{ (2,3), (2,4), (3,3)\}\) and use these problems for developing and testing a new elimination technique called elimination by degree steering.