This should be obvious, as I cannot seem to find further explanation anywhere on this definition. But first let me bring out the definition I am using.
where $X_1, Y_1 \in \frak{g_1}$ and $X_2, Y_2 \in \frak{g_2}$
Here what I don't get
So we wish to define a bracket on the direct sum, that is a bilinear map $$[,] : (\frak{g_1} \oplus \frak{g_2} \times \frak{g_1} \oplus \frak{g_2} \to \frak{g_1} \oplus \frak{g_2} )$$
Now vectors in the direct sum $\frak{g_1} \oplus \frak{g_2}$ take the form $X_1 + X_2$
Shouldn't we want $[X_1 + X_2, Y_1 + Y_2] = [X_1,Y_1] + [X_1, Y_2] + [X_2, Y_1] + [X_2, Y_2]$ by the bilinearity property?
Where do the $(,)$ pairs come from? And what is wrong with my definition?
Best Answer
The problem with your definition is that there is no obvious meaning to the terms $[X_1,Y_2]$ and $[X_2,Y_1]$. We can compute $[X_1,Y_1]$ using the bracket of $\mathfrak{g}_1$ and we can compute $[X_2,Y_2]$ using the bracket of $\mathfrak{g}_2$, but there's no way to compute $[X_1,Y_2]$ or $[X_2,Y_1]$.
The definition of the bracket on the direct sum gets around this problem by just declaring $[X_1,Y_2]=0$ and $[X_2,Y_1]=0$. That is, elements of $\mathfrak{g}_1$ commute with elements of $\mathfrak{g}_2$. As for the notation, $X_1+X_2$ and $(X_1,X_2)$ are different notations for the same thing. The latter notation refers literally to an element of the Cartesian product $\mathfrak{g}_1\times\mathfrak{g}_2$, which is the underlying set of $\mathfrak{g}_1\oplus\mathfrak{g}_2$. The former notation identifies $X_1\in\mathfrak{g}_1$ with $(X_1,0)$ and $X_2\in\mathfrak{g}_2$ with $(0,X_2)$, so $(X_1,X_2)$ is then the sum $X_1+X_2$.
So, when we declare that $[X_2,Y_1]=[X_1,Y_2]=0$, we get $$[X_1+X_2,Y_1+Y_2]=[X_1,Y_1]+[X_2,Y_2]$$ which just another name for the ordered pair $([X_1,Y_1],[X_2,Y_2])\in\mathfrak{g}_1\times\mathfrak{g}_2$.