#This shows that the Weyl group depends on the covering - SL(2,R) and
#PGL(2,R) are different

#First SL(2,R):
This is the Atlas of Reductive Lie Groups Software Package version 0.2.3.
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empty: type
Lie type: A1
elements of finite order in the center of the simply connected group:
Z/2
enter kernel generators, one per line
(ad for adjoint, ? to abort):
sc
enter inner class(es): s
main: realweyl
(weak) real forms are:
0: su(2)
1: sl(2,R)
enter your choice: 1
cartan class (one of 0,1): 0
Name an output file (hit return for stdout):

real weyl group is W^C.((A.W_ic) x W^R), where:
W^C is trivial
A is trivial
W_ic is trivial
W^R is trivial

#Now PGL(2,R):

real: type
Lie type: A1
elements of finite order in the center of the simply connected group:
Z/2
enter kernel generators, one per line
(ad for adjoint, ? to abort):
ad
enter inner class(es): s
main: realweyl
(weak) real forms are:
0: su(2)
1: sl(2,R)
enter your choice: 1
cartan class (one of 0,1): 0
Name an output file (hit return for stdout):

real weyl group is W^C.((A.W_ic) x W^R), where:
W^C is trivial
A is an elementary abelian 2-group of rank 1
W_ic is trivial
W^R is trivial

generators for A:
1