Test rank of free Lie algebras of the form F/R

n this thesis we define Magnus Embedding for free Lie algebras and mention briefly some properties of Fox derivatives. Afterwards, we calculate the test rank of some free Lie algebras. Let F be a free Lie algebra of rank n>=2 and R be an ideal of F. We show that an endomorphism of the algebra F/R?, which acts identically on R/R?, is an inner automorphism induced by some element of R/R?. Based on this result, we show that the test rank of the Lie algebra F/gama_k(F)? is n-1. After that we show the test rank of the Lie algebra F/R? is n-1 while R is an ideal of F such that F/R is free polynilpotent Lie algebra. Morever, we show the Lie algebra F/[R?,F] have test elements while rank is 2. In addition, we calculate the test rank of the abelyen product and finally, we show that the test tank of a free nilpotent Lie algebra is 1 or 2.