My question comes from the proof for proposition on page 127 of Fulton and Harris' Representation Theory, a first course. The proposition and its proof looks like this:
Also I cannot find any theorem relating to this one on Google, so I wonder if this proposition is even true.
Sorry for the formatting, I haven't figured out how to latex in post yet.
If $\newcommand{\g}{\mathfrak{g}}\g$ is a Lie algebra over a field $k$, then a one-dimensional representation is a Lie algebra map from $\g$ to $k$ considered as a Lie algebra. This is a linear map, but must preserve the Lie bracket. As the Lie bracket is zero on $k$ this means that it takes $[\g,\g]$ to zero. So a the one-dimensional representations correspond to linear maps $\g\to k$ taking $[\g,\g]$ to zero, equivalently to linear maps $\g/[\g,\g]\to k$.
In this example, one has $\newcommand{\la}{\lambda}\la:\text{Rad}(\g)\to k$ which vanishes on $\text{Rad}(\g)\cap [\g,\g]$. So we can extend $\la$ to $\text{Rad}(\g)+[\g,\g]$ by setting it to be zero on $[\g,\g]$. Then we can extend $\la$ to $\g$.
