Index 2 subgroups of Lie groups

Hi,

I know SO(2) is an index 2 subgroup of O(2). Are there any other Lie groups
that have index 2 subgroups? Is SO(n) also an index 2 subgroup of O(n)? (I
know O(2) is generated by rotations and reflections, and SO(2) is generated
by rotations, which is where we get index 2, but is this true in general for
O(n) and SO(n)?)
What about other Lie groups with index 2 subgroups other than orthogonal
groups?

Thank you,

James


.

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