When I am reading the Notes on Lie algebra by Hans Samelson, there is a sentence:
The standard skew-symmetric (exterior) form $det[X, Y ] = x_1y_2−x_2y_1$ on $\mathbb{C}^2$ is invariant under $sl(2,\mathbb{C})$ (precisely because of the vanishing of the trace).
Here, $sl(2,\mathbb{C})$ is the Special linear Lie algebra.
Can someone explain it?
Does it mean that for any $M \in sl(2,\mathbb{C})$, we have $det[MX, MY ] = det[X, Y ]$? But when $M=0$, it fails.
Thanks in advance.