The bi-invariant Haar measure on the quotient $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathrm{SL}_2(\mathbb{R}) / \mathrm{SO}_2(\mathbb{R})$ represents the moduli space of rank two real lattices modulo rotation and is easy to write down. An element of the quotient can be written (using the Isawasa decomposition) as: \[\begin{pmatrix} y^{1/2} & xy^{-1/2} \\ 0 & y^{-1/2} \end{pmatrix}\] with $z=x+iy$ in a fundamental domain of the action of $\mathrm{SL}_2(\mathbb{Z})$ over the upper-half plane, such as $D = \{(x,y) : x^2+y^2 \geq 1,|x| \leq 1/2,y > 0\}$. In these coordinates, the invariant measure is a scale of the hyperbolic measure, namely \[\frac{3}{\pi} \frac{dx \, dy}{y^2}.\] Is there a similar nice parametrisation for the rank 3 case (namely explicating the bi-invariant Haar measure of $\mathrm{SL}_3(\mathbb{Z}) \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}_3(\mathbb{R})$?

The bi-invariant Haar measure on the quotient $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathrm{SL}_2(\mathbb{R}) / \mathrm{SO}_2(\mathbb{R})$ represents the moduli space of rank two real lattices modulo rotation and is easy to write down. An element of the quotient can be written (using the Iwasawa decomposition) as: \[\begin{pmatrix} y^{1/2} & xy^{-1/2} \\ 0 & y^{-1/2} \end{pmatrix}\] with $z=x+iy$ in a fundamental domain of the action of $\mathrm{SL}_2(\mathbb{Z})$ over the upper-half plane, such as $D = \{(x,y) : x^2+y^2 \geq 1,|x| \leq 1/2,y > 0\}$. In these coordinates, the invariant measure is a scale of the hyperbolic measure, namely \[\frac{3}{\pi} \frac{dx \, dy}{y^2}.\] Is there a similar nice parametrisation for the rank 3 case (namely explicating the bi-invariant Haar measure of $\mathrm{SL}_3(\mathbb{Z}) \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}_3(\mathbb{R})$?

The bi-invariant Haar measure on the quotient $SL(2,\mathbb{R})/SL(2,\mathbb{Z})/SO(2,\mathbb{R})$ represents the moduli space of rank two real lattices modulo rotation and is easy to write down. An element of the quotient can be written (using the Isawasa decomposition) as: $$ \left(\begin{array}{cc} y^{1/2}&xy^{-1/2}\\ 0&y^{-1/2}\\ \end{array} \right) $$ with $z=x+iy$ in a fundamental domain of the action of $SL(2,\mathbb{Z})$ over the upper-half plane, such as $D = \{(x,y) : x^2+y^2 \geq 1,|x| \leq 1/2,y > 0\}$. In these coordinates, the invariant measure is a scale of the hyperbolic measure, namely $$ \frac{3dx dy}{\pi y^2}. $$ Is there a similar nice parametrisation for the rank 3 case (namely explicating the bi-invariant Haar measure of $SL(3,R)/SL(3,Z)/SO(3,\mathbb{R})$?

The bi-invariant Haar measure on the quotient $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathrm{SL}_2(\mathbb{R}) / \mathrm{SO}_2(\mathbb{R})$ represents the moduli space of rank two real lattices modulo rotation and is easy to write down. An element of the quotient can be written (using the Isawasa decomposition) as: \[\begin{pmatrix} y^{1/2} & xy^{-1/2} \\ 0 & y^{-1/2} \end{pmatrix}\] with $z=x+iy$ in a fundamental domain of the action of $\mathrm{SL}_2(\mathbb{Z})$ over the upper-half plane, such as $D = \{(x,y) : x^2+y^2 \geq 1,|x| \leq 1/2,y > 0\}$. In these coordinates, the invariant measure is a scale of the hyperbolic measure, namely \[\frac{3}{\pi} \frac{dx \, dy}{y^2}.\] Is there a similar nice parametrisation for the rank 3 case (namely explicating the bi-invariant Haar measure of $\mathrm{SL}_3(\mathbb{Z}) \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}_3(\mathbb{R})$?