  Suppose $G$ is a reductive algebraic group, $T$ is a Cartan subgroup,
$N=\text{Norm}(T)$, and $W=N/T$ is the Weyl group. If $w\in W$ has order $d$,
it is natural to ask about the orders lifts of $w$ to $N$. It is
straightforward to see that the minimal order of a lift of $w$ has order $d$ or
$2d$, but it can be a subtle question which holds. We first consider the
question of when $W$ itself lifts to a subgroup of $N$ (in which case every
element of $W$ lifts to an element of $N$ of the same order). We then consider
two natural classes of elements: regular and elliptic. In the latter case all
lifts of $w$ are conjugate, and therefore have the same order. We also consider
the twisted case.
