one-parameter subgroup
Pick one direction X in the Lie algebra and just keep moving that way. The path you trace, t -> exp(tX), is a one-parameter subgroup: a smooth curve through the identity that is also a homomorphism from the real line into the group. It is the cleanest picture of a continuous symmetry unfolding over time.
Formally a one-parameter subgroup is a smooth map gamma : R -> G with gamma(0) = I and gamma(s + t) = gamma(s) gamma(t) for all s, t. The homomorphism law says 'flowing for time s then time t equals flowing for time s + t'. A theorem: every such curve in a matrix Lie group has the form gamma(t) = exp(tX) for a unique X in the Lie algebra, namely X = gamma'(0).
So one-parameter subgroups and Lie algebra elements are the SAME data viewed two ways: X is the initial velocity, exp(tX) is the trajectory it generates. This is exactly the flow of the linear ODE x'(t) = X x(t): the algebra element X is the velocity field, the subgroup exp(tX) is the time-t flow map.
Why it matters: physical conservation laws and continuous symmetries (rotation in time, translation, phase evolution in quantum mechanics) all ARE one-parameter subgroups. A caveat: the image can wrap around. For irrational slopes on a torus the curve never closes and fills a dense subset, showing a one-parameter subgroup need not be a closed (embedded) submanifold.
A single generator X spins out the entire rotation group SO(2) as t runs over the reals — the canonical one-parameter subgroup.
The homomorphism law gamma(s+t) = gamma(s)gamma(t) is exactly the exponential law e^{s+t} = e^s e^t lifted to matrices — and it works precisely because tX and sX commute, so the only matrices being exponentiated together are scalar multiples of one X.