$\require{AMScd}$For a Riemannian manifold $M$, I have read authors talking about a 'Liouville measure' on the unit tangent bundle $\operatorname{T}^1(M)$ and then proceed to claim/prove that it is invariant under the geodesic flow. See for example Mautner's $1957$ paper 'Geodesic Flows on Symmetric Riemann Spaces.'

This measure is loosely defined as '*the measure on $\operatorname{T}^1(M)$ locally defined by $\omega \wedge \theta$ where $\omega$ is the volume form on $M$ associated to the metric and $\theta$ is an invariant volume form on the sphere*' (again, see equation ($17$) of Mautner's paper).

I would like a precise definition/construction of such a measure on a general symmetric space. I have seen the example of the upper half-plane where the unit tangent bundle can be identified with a group and one uses the Haar measure.

**Setup:**
Assume $(G,\mathfrak{g})$ is a non-compact, real, semi-simple, Lie group with finite center.
Fix a Cartan involution $\sigma: G \to G, d\sigma: \mathfrak{g} \to \mathfrak{g}$ which then induces the decomposition $\mathfrak{g}= \mathfrak{k}\oplus\mathfrak{p}$ into the $\pm 1$ eigenspaces respectively.
Moreover, if $K = G_{\sigma}$ (the fixed point set of $\sigma$), then $\operatorname{Lie}(K)= \mathfrak{k}$ and $K$ is compact since we assumed $G$ to have finite center.
One can also check that $\mathfrak{k}$ and $\mathfrak{p}$ are invariant under $\operatorname{Ad}(K)$.
By giving $\mathfrak{p}$ an $\operatorname{Ad}(K)$-invariant inner product, one can induce a $G$-invariant Riemannian metric on the tangent bundle of $G/K$.
Additionally, $G/K$ has Riemannian geodesics of form $t\mapsto g\exp(tX)K$, with $X\in \mathfrak{p}$.
Perhaps the example to keep in mind is $\operatorname{SL}_n(\mathbb{R}) \simeq \exp(\mathfrak{p}) \times \operatorname{SO}(n)$ where $\mathfrak{p}$ are symmetric, trace-zero matrices.

**What I want:** I would like to define a non-vanishing top form (Liouville measure) on $\operatorname{T}^1(G/K)$ which is 'locally of the type $\omega \wedge \theta$', where $\omega$ is the Riemannian volume form and $\theta$ is a non-vanishing top form on the sphere.
Note that $l_x$, left multiplication by $x\in G$, is an isometry on $G/K$.
And, of course, I would like the Liouville form to be invariant under the maps $dl_x$ so as to pass to the interesting class spaces $\operatorname{T}^1(\Gamma\backslash(G/K)) \simeq \Gamma\backslash \operatorname{T}^1(G/K)$, where $\Gamma$ is taken to be a freely acting lattice.
Lastly, I would like to know if there is some sort of uniqueness statement that can be formulated.

**An attempt:** A naive attempt would be to say that $\operatorname{T}^1(G/K) \simeq G/K \times S^d$ is a trivial sphere-bundle since $G/K\simeq \mathfrak{p}$ is contractible.
But I don't know how the action of $G$ would translate to $G/K \times S^d$.

Another attempt would be to use the commutative diagram

$$ \begin{CD} G\times \mathfrak{p}_1 @>>> G \times_{K} \mathfrak{p}_1 \\ @V{\pi_1}VV @VV{\overline{q\circ\pi_1}}V \\ G @>{q}>> G/K \end{CD}$$

Here $\mathfrak{p}_1$ are the vectors in $\mathfrak{p}$ of norm $1$, $K$ acts on $G\times \mathfrak{p}_1$ on the right by $$(g,Y)\cdot k := (gk,\operatorname{Ad}(k^{-1})Y),$$ and $G \times_{K} \mathfrak{p}_1$ is shorthand for the quotient by this action. Note that $G \times_{K} \mathfrak{p}_1$ can be identified with $\operatorname{T}^1(G/K)$ by using the map $$(g,Y) \mapsto (g\exp(tY)K)'_{t=0}.$$ Under this identification, the action of $dl_x: \operatorname{T}^1(G/K) \to \operatorname{T}^1(G/K)$ for $x\in G$ is intertwined with the action $x\cdot(g,Y)K := (xg,Y)K$ on the first coordinate.

Thus it perhaps makes sense to try and construct a form $\alpha$ (of appropriate dimension) on $G\times \mathfrak{p}_1$ which is invariant (under both the left action of $G$ on the first component and the right action of $K$ specified above) and is also $K$-horizontal, that is to say that the interior product $\iota_X \alpha =0$ whenever $X$ is a vector field induced by the action of $K$. I guess such a form would induce a top form on $G\times_K \mathfrak{p}_1$ with the required properties?

Any help/references would be appreciated.