The exponential map for the unit circle is $it \to \exp (it) = \cos t + i \sin t$
This group can be augmented with a norm on the complex plane such that $r$ is the length of the complex number on the plane: $r\exp(it)$. Specifically, where $(a,b)$ is a position on the complex plane, the length is given by $r=\sqrt{a^2+b^2}$
This seems like it should be true for any Lie algebra; one can always multiply the element by a scalar and attribute a certain length to it.
For instance , the special linear group has a map $Xt \to \exp (Xt)$. If I assign a length to the special linear group, I also get a length $\exp(a+Xt)=r\exp(Xt)$, which should now be the general linear group.
But, whereas in the case of the complex number the norm was $r=\sqrt{a^2+b^2}$, I am at a total loss as to what the expression of the length would be for other algebras. Is the choice arbitrary, or should I take the length of, say the su(2) Lie algebra, to be $r=\sqrt{x^2+y^2+z^2}$ under the mapping
$$ \ln r+x\sigma_1+y\sigma_2+z\sigma_3\to r\exp(x\sigma_1+y\sigma_2+z\sigma_3) $$
Now, I have randomly guessed $r$ to be $r=\sqrt{x^2+y^2+z^2}$, but how do I know its the right formulation. Are there rules to follow?
Is the a special type of plane for each Lie algebra, analogous to the complex plane for the complex numbers, where I can place its element, such that a certain norm is better than the others?