The question is about the matrix representation of the following 6-dimensional Lie algebra, with 6 generators $t_1,t_2,t_3,t_4,t_5,t_6$. This Lie algebra is nilpotent, non-abelian, non-reductive and solvable. The commutation relations between all the generators are:
$$[t_1,t_2]=t_6,$$ $$[t_2,t_3]=t_4,$$ $$[t_3,t_1]=t_5,$$ while all the other commutators are zeros.
The invariants of this Lie algebra is found to be $t_4,t_5,t_6$ and $t_1t_4+t_2t_5+t_3t_6$.
Questions: What are the allowed matrix representation of the above 6-dimensional Lie algebra? What are the rank of the matrices of the representation?
The Ref is TABLE III in Invariants of real low dimension Lie algebras J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus Citation: Journal of Mathematical Physics 17, 986 (1976); doi: 10.1063/1.522992. I had switch the notations from $e_1 \to t_1,e_2 \to t_2,e_3 \to t_3, e_5 \to t_4,e_4 \to -t_5,e_6 \to t_6$ for a more symmetric form.
