Let F denote an algebraically closed field. Denote the
three-element set by X={A,B,C}, and let
\mathbb{F}\left denote the free unital associative
F-algebra on X. Fix a nonzero qβF such that
q4ξ =1. The universal Askey-Wilson algebra Ξ is the quotient space
\mathbb{F}\left/\mathbb{I}, where I is the
two-sided ideal of \mathbb{F}\left generated by the nine
elements UVβVU, where U is one of A,B,C, and V is one of
\begin{equation} (q+q^{-1}) A+\frac{qBC-q^{-1}CB}{q-q^{-1}},\nonumber
\end{equation} \begin{equation} (q+q^{-1})
B+\frac{qCA-q^{-1}AC}{q-q^{-1}},\nonumber \end{equation} \begin{equation}
(q+q^{-1}) C+\frac{qAB-q^{-1}BA}{q-q^{-1}}.\nonumber \end{equation} Turn
\mathbb{F}\left into a Lie algebra with Lie bracket
[X,Y]=XYβYX for all X,Y\in\mathbb{F}\left.
Let L denote the Lie subalgebra of
\mathbb{F}\left generated by X, which is also
the free Lie algebra on X. Let L denote the Lie subalgebra of
Ξ generated by A,B,C. Since the given set of defining relations of
Ξ are not in L, it is natural to conjecture that L is
freely generated by A,B,C. We give an answer in the negative by showing that
the kernel of the canonical map
\mathbb{F}\left\rightarrow\Delta has a nonzero
intersection with L. Denote the span of all Hall basis elements of
L of length n by Lnβ, and denote the image of
βi=1nβLiβ under the canonical map LβL
by Lnβ. We study some properties of L4β and L5β