Past exam of the lie algebras course of Part C of the mathematics course of the University of Oxford / 2023 / 1 / c / i

First put the nonabelian two-dimensional Lie algebra in a convenient form. For any basis $p,q$, let $u=[p,q]=ap+bq\ne 0$. Then $[p,u]=bu$ and $[q,u]=-au$. Since $(a,b)\ne (0,0)$, some linear combination $x$ of $p,q$ has $[x,u]=u$. In particular, $x,u$ are linearly independent. Taking $y=u$ gives a basis withThis is the affine Lie algebra of the line.

Every alternating bilinear form on this vector space isFor any alternating bilinear form $\alpha$, the expressionis an alternating trilinear form: it is trilinear, and setting any two arguments equal makes the terms cancel. Every alternating trilinear form on a two-dimensional vector space vanishes, since any triple is linearly dependent. Hence every ${\alpha }_c$ satisfies the Lie algebra two-cocycle condition, and there are no others: