I'm reading "Ergodic Theory with a view towards Number Theory" and I met the same problem discussed here.
The book shows how to construct a metric on a general Lie group but it doesn't mention the metric on $SL_2(\mathbb{R})$.
I need to show that the Left-Invariant Riemannian Metric discussed in Chapter 9 is equivalent to the Euclidean metric induced by the immersion of $SL_2(\mathbb{R})$ in $\mathbb{R}^4$ (Frobenius Norm) without knowing too much of the Riemannian Metric.
I approched the problem in the following way.
Let $d$ the Left-Invariant Riemannian Metric and $d_{F}$ the metric induced by the Frobenious norm on $M_{2}(\mathbb{R})$ defined as follow:
$$
\begin{align}
d_{F}(g_{1},g_{2})=\|g_{1}-g_{2}\|_{F}
=\sqrt{ \mathrm{tr}((g_{1}-g_{2})^{T}(g_{1}-g_{2})) } \\
\end{align}
$$
I want to show that they are equivalent:
$$
\exists c>0 \ c d_{F}(g_{1},g_{2})\leq d(g_{1},g_{2}) \leq\frac{1}{c}d_{F}(g_{1},g_{2})\quad \forall g_{1},g_{2}\in SL_{2}(\mathbb{R})
$$
Suppose holds
$$
\exists c>0 \ c d_{F}(I,g)\leq d(I,g) \leq\frac{1}{c}d_{F}(I,g)\quad \forall g\in SL_{2}(\mathbb{R})\tag{1}
$$
By the left-invariancy we have $d(g_{1},g_{2}) = d(I,g_{1}^{-1}g_{2})$ for each $g_{1},g_{2}\in SL_{2}(\mathbb{R})$.
We have to show
$$
\begin{align}
\exists c_{1} >0\ d_{F}(g_{1},g_{2})\leq c_{1} d_{F}(I,g_{1}^{-1}g_{2}) \\
\exists c_{2} >0\ d_{F}(I,g_{1}^{-1}g_{2})\leq c_{2}d_{F}(g_{1},g_{2})
\end{align}
$$
$$
\begin{align}
d_{F}(g_{1},g_{2})&=\|g_{1}(I-g_{1}^{-1}g_{2})\|_{F} \\
&\leq\|g_{1}\|_{F}\|I-g_{1}^{-1}g_{2}\|_{F} && \text{submultiplicativity Frobenious}\\
&=\|g_{1}\|_{F}d_{F}(I,g_{1}^{-1}g_{2})
\end{align}
$$
$$
\begin{align}
d_{F}(I,g_{1}^{-1}g_{2})&=\|I-g_{1}^{-1}g_{2}\|_{F} \\
&\leq\|g_{1}^{-1}\|_{F}\|g_{1}-g_{2}\|_{F} && \text{submultiplicativity Frobenious}\\
&=\|g_{1}^{-1}\|_{F}d_{F}(g_{1},g_{2})
\end{align}
$$
It remains to prove $(1)$, how could I do that without knowing explicitly the metric?
I really appreciate any kind of help.