Analysis is often easier in terms of these generators, rather than the full rotation matrix. Analysis in terms of the generators is known as the [[Lie algebra]] of the rotation group.
=== Relationship with Lie algebras ===
The matrices in the [[Lie algebra]] are not themselves rotations; the skew-symmetric matrices are derivatives, proportional differences of rotations. An actual "differential rotation", or ''infinitesimal rotation matrix'' has the form
:<math> I + A \, d\theta ~,</math>
where {{math|''dθ''}} is vanishingly small and {{math|''A'' ∈ '''so'''(n)}}, for instance with {{math|1=''A'' = ''L''<sub>''x''</sub>}},
The computation rules are as usual except that infinitesimals of second order are routinely dropped. With these rules, these matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals.<ref>{{Harv|Goldstein|Poole|Safko|2002|loc=§4.8}}</ref> It turns out that ''the order in which infinitesimal rotations are applied is irrelevant''. To see this exemplified, consult [[Rotation group SO(3)#Infinitesimal rotations|infinitesimal rotations SO(3)]].
