What does it mean that "all Cartan subalgebras of a semisimple Lie algebra are conjugates"? I know this refers to adjoint action but I don't know exactly what it means.

The most obvious definition to me is that if $\mathfrak{h}\_1$ and $\mathfrak{h}\_2$ are two Cartan subalgebras, then there exists some $x \in \mathfrak{g}$ such that $[x,\mathfrak{h}\_1] = \mathfrak{h}\_2$. However I don't see why $[x,\mathfrak{h}\_1]$ is even a subalgebra at all, so maybe this doesn't make sense.