In Duistermaat's and Kolk's $\textit{Lie Groups}$ on page 2, they say that for a Lie group $G$ with $\mathfrak{g}=T_1G$ we can use the fact that $x\rightarrow xy$ and $y\rightarrow xy$ are identity maps if $y=1$ and $x=1$ respectively and they combine this with: $$\left (\begin{matrix} X \\ Y \end{matrix}\right ) = \left (\begin{matrix} X \\ 0 \end{matrix}\right ) + \left(\begin{matrix} 0 \\ Y \end{matrix}\right)$$ to obtain: $$T_{1,1}\mu:(X,Y)\rightarrow X+Y:\mathfrak{g}\times\mathfrak{g}\rightarrow \mathfrak{g}$$
$\textbf{Question:}$ How did this happen? I don't know what they are doing here. Why are they using those identity maps in the beginning and how does it relate to what seems to be written down as COLUMN VECTORS??? They don't even say what $X$ and $Y$ are although from context it seems to belong to $\mathfrak{g}$.
They go further on to say:
Applying this to $x \rightarrow xi(x)\equiv 1$ and using the sum rule for differentiation gives: $$T_1 i: X\rightarrow -X: \mathfrak{g}\rightarrow \mathfrak{g}$$ so these two mappings define an additive group structure on $\mathfrak{g}$ making $\mathfrak{g}$ itself a commutative Lie group.
$\textbf{Question:}$ How does the sum rule for differentiation come in at this last part? I don't see it.
P.S: This is the $\textbf{only}$ place where I've seen any mention of a Lie algebra being treated as a Lie group in itself in this manner. No other source seems to mention it at all.
A Lie algebra is a vector space (plus some more structure) and as such it also has a group structure via addition of vectors. Addition is also continuous so any (finite dimensional) vector space is a Lie group, just not a very exciting one.
What he seems to be doing in the beginning is showing that if you want to define a Lie group structure on a Lie algebra that is compatible with the Lie group, the trivial one is the natural choice to make.