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 $g$.