The algorithms that recognize Milnor laws and properties of these laws

We consider several equivalent definitions of the so-called Milnor laws (or Milnor identities) that is the laws which are not satisfied in $\mathfrak{A}_p\mathfrak{A}$ varieties. The purpose of this article is to provide algorithms that allow us to check whether a given identity $w(x,y)$ has one of the following properties:

• $w(x,y)$ is a Milnor law,
  
• every nilpotent group satisfying $w(x,y)$ is abelian,
  
• every finitely generated metabelian group satisfying $w(x,y)$ is finite-by-abelian.