Concrete example of Lie algebra in which a composition $X\cdot Y$ is not in the Lie algebra?

Question

I'm currently reading Fulton and Harris' Representation Theory text. In particular, I am looking at Chapter 8 where they introduce the definition of the Lie bracket. A question they raise is: Why could we not simply define the Lie bracket to be $$[X,Y] = XY - YX?$$ Why did we have to consider the maps Ad and ad?

The reason for this is that for any embedding of a Lie group $G$ into a general linear group $GL(V)$, we have a corresponding embedding of its Lie algebra $\mathfrak{g}$ into $\text{End}(V)$. If we simply defined $[X,Y] = XY - YX$, we need to define what is meant by the composition $XY$ and $YX$. These however depend on the embedding and may not even be an element of the Lie algebra.

Can someone provide a concrete example of when the composition $X \cdot Y$ may be not even be an element of the Lie algebra?

Answer 1

For example, let $\mathfrak{g}$ consists of matrices in $\text{End}(V)$ of trace zero. But the product of two matrices of trace zero might not be of trace zero.

Answer 2

There are multiple questions.

1. Why could we not simply define the Lie bracket to be $[X,Y] = XY - YX?$

Answer: In an abstract Lie algebra $L$, there is no product $XY$ defined for $X,Y\in L$, only a Lie bracket $[X,Y]$. The elements of $L$ are not necessarily matrices.

2. Can someone provide a concrete example of when the composition $X \cdot Y$ may be not even be an element of the Lie algebra?

Yes, take $L=\mathfrak{sl}_n(K)$, the Lie algebra of trace zero matrices of size $n$.