Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 302 / 1 / a / Solution

Put $\Lambda (t)=I+tX+O(t^2)$ in ${\Lambda }^T\eta \Lambda =\eta$. The coefficient of $t$ is $X^T\eta +\eta X$, so it must vanish. For $X=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ this condition isHence $a=d=0$ and $b=c$. Thus the Lie algebra is one-dimensional with basis