How is Lie theory relevant to robotics and optimisation? I looked into Weyl groups a bit during my undergrad, but didn't really go into Lie theory in much depth.
There are other cool things like representing a probability distribution over a 3D rotation with an multivariate gaussian vector interpreted as a so3 or se3 vector. Using this you can for implement a very elegant extended kalman filter estimating a 3D rotation. Which usually can be a tricky thing to do.
In robotics we pretty much only use the SO3, SE3 and SIM3 groups so my knowledge is limited to those groups.
A 3 valued so3 lie algebra vector is however always a valid 3D rotation, no matter what the values are. That makes it an ideal parameterization of the optimization problem.
The mapping from so3 lie algebra to SO3 lie group (which could be represented as a rotation matrix or quaternion) is a smooth differentiable function, which allows us to compute the jacobian we need for the gradient in the optimization.