# A Wilson line property

Consider a path, say $P$, and three points on it like $x,y,z$. If there were infinitesmally closed, then the following relation would be true.

$$W_{P}(x,y) W_{P}(y,z)=W_{P}(x,z)$$

If the Lie algebra of the theory was Abelian, it was also held for any three points on the path no matter how far they are. Is it also true in non-Abelian cases? The Wilson line of the path $P$ contains path-ordering so I am not sure whether this is the case in non-Abelian Lie algebras.

Yes, non-Abelian Wilson lines $W_P[\gamma]$ form a groupoid via concatenation of oriented curves and has OP's mentioned property by definition of path-ordering of ordered exponentials.