A left-invariant Riemannian metric on Lie group is a special case of homogeneous Riemannian manifold, and its differential geometry (geodesics and curvature) can be described in a quite compact form. I am most familiar with the description in 28.2 and 28.3 of here of covariant derivative and curvature.
But on a Lie group itself there is an explicit description of Jacobi fields available for
right invariant metrics (even on infinite dimensional Lie groups) in section 3 of:
- Peter W. Michor: Some Geometric Evolution Equations Arising as Geodesic Equations on Groups of Diffeomorphism, Including the Hamiltonian Approach. IN: Phase space analysis of Partial Differential Equations. Series: Progress in Non Linear Differential Equations and Their Applications, Vol. 69. Birkhauser Verlag 2006. Pages 133-215. (pdf).
I shall now describe the results (which go back to Milnor and Arnold). For detailed computations, see the paper.
Let $G$ be a Lie group with Lie algebra
$\def\g{\mathfrak g}\g$.
Let $\def\x{\times}\mu:G\x G\to G$ be the multiplication, let $\mu_x$ be left
translation and $\mu^y$ be right translation,
given by $\mu_x(y)=\mu^y(x)=xy=\mu(x,y)$.
Let $\langle \;,\;\rangle:\g\x\g\to\Bbb R$ be a positive
definite inner product. Then
$$\def\i{^{-1}}
G_x(\xi,\eta)=\langle T(\mu^{x\i})\cdot\xi,
T(\mu^{x\i})\cdot\eta)\rangle
$$
is a right invariant Riemannian metric on $G$, and any
right invariant Riemannian metric is of this form, for
some $\langle \;,\;\rangle$.
Let $g:[a,b]\to G$ be a smooth curve.
In terms of the right logarithmic derivative $u:[a,b]\to \g$ of $g:[a,b]\to G$, given by
$u(t):= T_{g(t)}(\mu^{g(t)\i}) g_t(t)$,
the geodesic equation has the expression
$$ \def\ad{\text{ad}}
\partial_t u = u_t = - \ad(u)^{\top}u\,,
$$
where $\ad(X)^{\top}:\g\to\g$ is the adjoint of $\ad(X)$ with respect to the inner product $\langle \;,\; \rangle$ on $\g$, i.e.,
$\langle \ad(X)^\top Y,Z\rangle = \langle Y, [X,Z]\rangle$.
A curve $y:[a,b]\to \g$ is the right trivialized version of a Jacobi field along the geodesic $g(t)$ described by $u(t)$ as above iff
$$
y_{tt}= [\ad(y)^\top+\ad(y),\ad(u)^\top]u
- \ad(u)^\top y_t -\ad(y_t)^\top u + \ad(u)y_t\,.
$$
Continued:
For connected $G$, the right invariant metric is biinvariant iff $\ad(X)^\top = -\ad(X)$.
Then the geodesic equation and the Jacobi equation reduces to
$$
u_t = \ad(u)u = 0,\qquad y_{tt} = \ad(u)y_t
$$
Now we can look at examples.
If $G=SU(2)$ then $\g=\mathfrak{sl}(3,\mathbb R)$ and we can take an arbitrary inner product on it.
(Maybe, I will continue if I have more time in the next few days).
Best Answer
Consider the fiber bundle \begin{align} K\rightarrow G\rightarrow G/K. \end{align}
Herein we assume $G$ is connected.
Since $G/K$ is non-positively curved, it is aspherical (i.e. all its higher homotopy groups vanish). Thus, the long exact sequence of homotopy groups induced by the above fiber bundle implies the short exact sequence \begin{align} 1\rightarrow \pi_1(K)\rightarrow \pi_1(G)\rightarrow \pi_1(G/K)\rightarrow 1. \end{align} The inclusion of $K\rightarrow G$ is a homotopy equivalence (this requires proof but is standard and for matrix groups comes from the polar decomposition), and hence the induced map \begin{align} \pi_1(K)\rightarrow \pi_1(G) \end{align} is an isomorphism. Hence, by the previous short exact sequence $\pi_1(G/K)=1$ and $G/K$ is simply connected.