  In a Lie group equipped with a left-invariant metric, we study the minimizing
properties of geodesics through the presence of conjugate points. We give
criteria for the existence of conjugate points along steady and nonsteady
geodesics, using different strategies in each case. We consider both general
Lie groups and quadratic Lie groups, where the metric in the Lie algebra
$g(u,v)=\langle u,\Lambda v\rangle$ is defined from a bi-invariant bilinear
form and a symmetric positive definite operator $\Lambda$. By way of
illustration, we apply our criteria to $SO(n)$ equipped with a generalized
version of the rigid body metric, and to Lie groups arising from Cheeger's
deformation technique, which include Zeitlin's $SU(3)$ model of hydrodynamics
on the $2$-sphere. Along the way we obtain formulas for the Ricci curvatures in
these examples, showing that conjugate points occur even in the presence of
some negative curvature.
