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 $$ y_{tt}= [\ad(y)^\top+\ad(y),\ad(u)^\top]u - \ad(u)^\top y_t -\ad(y_t)^\top u + \ad(u)y_t\,. $$ 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. (I will continue if I have more time in the next few days).

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).