Are there any known common parametrizations of SL(3,R)? I know that it is easy to obtain a local parametrization by just exponentiating generators from the Lie algebra, but I do not know if they are global or not. How can I know if a parametrization obtained by exponentiating Lie algebra generators is a global parametrization?


As a second part of this question, SL(2,R) can be easily parametrized with three variables $t,\phi, \chi$ as
$$\left[\begin{array}{cc}\cosh\chi\cos t+\sinh\chi\sin\phi&\sinh\chi\cos\phi+\cosh\chi\sin t\\\sinh\chi\cos\phi-\cosh\chi\sin t&\cosh\chi\cos t-\sinh\chi\sin\phi\end{array}\right]$$


Are there similar parametrizations for SL(3,R), that have reasonably simple expressions for the elements of the matrix? If I try to exponentiate the Lie algebra elements the resulting matrix elements are a mess.



Parametrization of SL(3,R)
By singular value decomposition, every matrix in $SL\_3$ factors as $UDV$ with $U$ and $V$ in $SO(3)$ and $D$ of the form $\mathrm{diag}(x,y,1/(xy))$ (or $\mathrm{diag}(e^u,e^v,e^{-u-v})$ if you prefer). Any matrix in $SO(3)$ can be factored as
$$\begin{bmatrix}
\cos \alpha & - \sin \alpha & 0 \\
\sin \alpha & \cos \alpha & 0 \\
0&0&1 \\
\end{bmatrix}
\begin{bmatrix}
\cos \beta & 0 & \sin \beta \\
0&1&0 \\
 -\sin \beta & 0 &\cos \beta \\
\end{bmatrix}
\begin{bmatrix}
1&0&0 \\
0 & \cos \gamma & -\sin \gamma \\
 0 & \sin \gamma &\cos \gamma \\
\end{bmatrix}$$
(Euler angle factorization). Is that the sort of formula you want?


