We now know what a Lie algebra is and we know they are abstractions of infinitesimal symmetries, which are given by derivations. Today we will see what we can say about associating infinitesimal symmetries to continuous symmetries: that is, given a matrix Lie group 
As in the other post, I will be ignoring some technical details for the sake of exposition. For example, I am generally not specifying how I’m topologizing various objects, and this is because of the general fact that a finite-dimensional real vector space has a unique Hausdorff topology compatible with addition and scalar multiplication. Whenever I talk about limits in such a vector space, I therefore don’t need to specify how I’m imposing a topology, although it will generally be convenient to induce it via a norm (which I am also not specifying).
. Part of the reason it is so simple to understand is that 
of its elements, showing that it can be understood either in terms of the group of elements of norm 
in 
(that is, the unitary group 
) or the imaginary subspace of 
-sphere 
. It turns out that there is a picture completely analogous to the picture above, but with 
: that is, 
is isomorphic to the group of elements of norm 
), and there is an exponential map from the imaginary subspace of 
lets us handle elements of 
almost as easily as we handle elements of 
. 
Annoying Precision 