I have heard that the second homotopy group of some Lie groups is trivial, but I am confused about the conditions under which this is true. In some references there are no conditions at all, in some others that it should be compact, while others assume that it should be connected as well. So, which is the minimum set of requirements under which the second fundamental group of a Lie group is trivial?
Any references that clarify this matter would be appriciated.
It's true. Connectedness doesn't matter, for $\pi_2$ is computed wrt a basepoint, and any part of the group not in the same component as the basepoint will have no effect.
Compactness makes it easier to prove, I believe. But see https://mathoverflow.net/questions/8957/homotopy-groups-of-lie-groups
which uses the (big) fact that a Lie group deformation retracts onto something compact, which is the only missing step.