I am a physicist and we do Lie algebras pretty informally, so I hope my question makes any sense to a mathematician. There is one thing that I don't quite understand, which is why we can find a basis for the elements of the Lie algebra?
Just to show you at what kind of informal level I understand the concepts so far, here is my line of thought: We have a group and find a set of matrices that form a representation for the group. Then for every such matrix $O$ we find a corresponding matrix in the Lie algebra $H$ such that: $O = e^{iH}$. Now we say that there we can find a basis for these elements, i.e. a finite set $H_i$ such that for any $H$ exist $\alpha_j$: $H=\sum_j \alpha_j H_j$. If anything so far is incorrect please let me know.
Now regarding my actual question: I thought to be able decompose elements like this would that not require the property that for 3 elements $O_1, O_2, O_3$ with $O_3 = O_1 O_2$ we also have
\begin{equation} \alpha_j^{(3)} = \alpha_j^{(1)}+\alpha_j^{(2)} \end{equation}
(where $\alpha_j^{(1)}$ are the Lie algebra coefficients of $O_1$ and similarly for 2, 3)? This is of course not true since the generators do not commute. What am I missing? Is there maybe a different relation between the coefficients of 3 group elements that are related by a multiplication other than adding them?