Pensieve header: A solvable algebra which is not a semi-direct product A+N where N is nilpotent, A is Abelian, and A acts on N semi-simply.

A={{1,0,0},{0,0,1},{0,0,0}};
X={{0,0,1},{0,0,0},{0,0,0}};
Y={{0,-1,0},{0,0,0},{0,0,0}};

MatrixForm/@{A,X,Y}

Comm[a_,b_]:=a.b-b.a

Comm[A,X]===X 
Comm[A,Y]===X+Y
Comm[X,Y]===0A

Permutations[{a,x,y}]

MatrixForm /@ ((p|->Dot@@(MatrixExp/@(p/.{a->α A, x->ξ X, y->η Y}))) /@ Permutations[{a,x,y}])