$ \DeclareMathOperator\SO{SO} \DeclareMathOperator\SU{SU}$ For which compact connected simple lie groups $ G $ does the sequence $$ 1 \to T \to N(T) \to W \to 1 $$ split? Here $ T $ is the maximal torus, $ N(T) $ is the normalizer of the maximal torus and $ W $ is the Weyl group.
Let $ T^r $ denote a torus of rank $ r $.
For $ SO(3) $ the extension splits $ N(T)=T^1:2=O(2,\mathbb{R}) $. I was assuming this extension would always split. But then I thought more about $ SU(2) $ and realized that $ N(T)=T^1 \cdot 2 $ is non-split for $ SU(2) $ since there is a unique element of order $ 2 $ and it is in the identity component.
This is Theorem 4.16 of https://arxiv.org/abs/1608.00510
Let $ G $ be a compact simple Lie group. Then $ N(T) $ splits as $ N(T)=T \rtimes W $ for the following cases:
$ PSU(n) $, and more generally any group $ G $ of type $ A_n $ with $ |Z(G)| $ odd. Also $ SU(4)/\pm I \cong SO(6) $, even though the center has size $ 2 $.
$ SO(2n+1) $
$ PSp(1) \cong SO(3) $ and $ PSp(2) \cong SO(5) $
$ SO(2n) $ and $ PSO(2n) $
$ G_2 $
And $ N(T) $ does not split for:
Groups $ G $ of type $ A_n $ with $ |Z(G)| $ even, except for $ SU(4)/\pm I \cong SO(6) $ where $ N(T) $ does split, see above
$ Spin(2n+1) $
$ PSp(n) $ for $ n \geq 3 $, and all $ Sp(n) $
$ Spin(2n) $
All other exceptional groups $ F_4, E_6, 3.E_6, E_7, 2.E_7, E_8 $
The reference also notes that if $ N(T) $ splits for some group $ G $ then $ N(T) $ must also split for the adjoint form of $ G $. Indeed note that $ N(T) $ splits for $ PSU(n), SO(2n+1),PSO(2n) $. And to the contrapositive we have that $ N(T) $ does not split for $ PSp(n), n \geq 3 $ and so indeed $ N(T) $ also fails to split for the corresponding $ Sp(n) $.