  The lowest two-sided cell of the extended affine Weyl group $W_e$ is the set
$\{w \in W_e: w = x \cdot w_0 \cdot z, \text{for some} x,z \in W_e\}$, denoted
$W_{(\nu)}$. We prove that for any $w \in W_{(\nu)}$, the canonical basis
element $\C_w$ can be expressed as $\frac{1}{[n]!} \chi_\lambda({\y}) \C_{v_1
w_0} \C_{w_0 v_2}$, where $\chi_\lambda({\y})$ is the character of the
irreducible representation of highest weight $\lambda$ in the Bernstein
generators, and $v_1$ and $v_2^{-1}$ are what we call primitive elements.
Primitive elements are naturally in bijection with elements of the finite Weyl
group $W_f \subseteq W_e$, thus this theorem gives an expression for any
$\C_w$, $w \in W_{(\nu)}$ in terms of only finitely many canonical basis
elements. After completing this paper, we realized that this result was first
proved by Xi in \cite{X}. The proof given here is significantly different and
somewhat longer than Xi's, however our proof has the advantage of being mostly
self-contained, while Xi's makes use of results of Lusztig from \cite{L
Jantzen} and Cells in affine Weyl groups I-IV and the positivity of
Kazhdan-Lusztig coefficients.
