From Lie groups to Lie algebras

Previously we have seen that infinitesimal rotation in three dimensions are generated by the matrices \(t_x\), \(t_y\) and \(t_z\) which obey the commutation relations
\[ \begin{equation}
 [t_x, t_y ] = t_z \\
 [t_y, t_z ] = t_x \\
 [t_z, t_x ] = t_y
\label{flg:eq:comrel1}
\end{equation}
\] It turns out that \(t_x\), \(t_y\) and \(t_z\) form the basis of a three-dimensional vector space \(\cal{L}\). The commutation relations \eqref{flg:eq:comrel1} imply however, that \(\cal{L}\) is more than an ordinary vector space. It has additional structure, viz. a "multiplication" \([\cdot,\cdot]\) which maps two elements of \(\cal{L}\) into an element of \(\cal{L}\). Formally, we write
\[ 
[a, b] = c
\] with \(a\), \(b\) and \(c \in \cal{L}\). The "multiplication" \([\cdot,\cdot]\) is anticommutative
\[
[a,b] = -[b,a]
\] and it obeys the Jacobi identity
\[
[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0
\] If a vector space is equipped with this type of "multiplication", it is called a Lie algebra.

If I understand correctly, we can already learn a lot about the Lie group \(L\) if we narrow our view to these infinitesimal rotations, i.e. the local neighborhood of \(L\)'s unit element and study the Lie algebra \(\cal{L}\) spanned by the generators \(t_i\), rather that the Lie group \(L\).

Restricting our investigation to the vicinity of \(L\)'s unit element is not a serious limitation, since any element \(g\) of the Lie group \(L\) can be turned into the unit element simply by multiplying all elements of \(L\) with \(g^{-1}\), the inverse element of \(g\).