Let $L$ be a free Lie algebra with basis $X$, and $A$ be its subalgebra wit basis $B$. Let $x \in X$ but not in $B$. Let also consider $t$ be an arbitrary element. How can we show that $t$ and $x$ are a free basis for the subalgebra generated by them? Indeed, I have a problem with freely generated subalgebra?
If algebra is not Schreier variety (every subalgebra of free algebra is also free), how can we define a free subalgebra of that?
Many thanks for your help.