Good morning everyone. I've encountered recently during my computations the following lie algebra


$$\mathfrak g=\text{span}(f\_0,f\_1,f\_2),$$
with $$\begin{eqnarray}[f\_2,f\_1]&=&f\_0+a f\_2,\\ [f\_1,f\_0]&=&bf\_2,\\ [f\_2,f\_0]&=&0,\qquad a,b\in \mathbb R\setminus\{0\}\end{eqnarray}$$


Of course $$\dim[\mathfrak g,\mathfrak g]=2,$$ and the algebra is solvable, moreover I know we can interpret it as the operator $$\text{ad}f\_1$$ acting on $$\text{span}(f\_0,f\_2).$$ I was wondering to know if there are known casimirs for this algebra or any suggestions or references on how to find them.


Thanks anyone for helping me.


Best wishes



Casimir of a three dimensional solvable lie algebra
Casimirs for low-dimensional Lie algebras are given explicitly in [Invariants of real low-dimensional Lie algebras](http://link.aip.org/link/doi/10.1063/1.522992) by J Patera, RT Sharp, P Winternitz and H Zassenhaus.


As you point out, the Lie algebra in question is the semidirect product of the two-dimensional abelian Lie algebra spanned by $(f\_0,f\_2)$ by the one-dimensional Lie algebra spanned by $f\_1$, where the action of $f\_1$ is given by the matrix
$$
A = \begin{pmatrix} 0 & 1 \cr b & a \end{pmatrix}
$$


Since $a\neq 0$, the matrix has nonzero trace and hence the Lie algebra is not unimodular. The characteristic polynomial of the above matrix is $\chi\_A(t) = t^2 - a t - b$. We have to to consider three different cases:


1. $b > -1/4 a^2$, in which case the Lie algebra is isomorphic to $A\_{3,5}$ in that paper,
2. $b=-1/4 a^2$, in which case it is isomorphic to $A\_{3,2}$, and
3. $b < -1/4 a^2$, in which case it is isomorphic to $A\_{3,7}$.


I leave you the exercise of relating your basis to the ones in the paper (hint: bring the matrix to normal form) so that you can read off the invariants from their Table I.


