Pensieve header: A solvable Lie algebra which is not a semi-direct product of an Abelian and a nilpotent.

Clear[e,A,B,X,Y,Z,Comm];

e[i_,j_]:=SparseArray[{{i,j}->1},{7,7}]//Normal;

A=e[1,2]+e[4,4];
B=e[2,3]+e[6,6];

X=e[4,5];
Y=e[6,7];
Z=e[1,3];

Comm[u_,v_]:=u.v-v.u;

{Comm[A,B]==Z,Comm[A,X]==X,Comm[B,Y]==Y,Comm[A,Y]==0 Y,Comm[B,X]==0 X,Comm[A,Z]==0 Z,Comm[B,Z]==0 Z,Comm[X,Y]==0 X,Comm[X,Z]==0 Z,Comm[Y,Z]==0 Z}

Convex basis: azbxy.

AllExps=((p|->(p->Dot@@(MatrixExp/@(p/.{a->α A,b->β B, x->ξ X, y->η Y,z->ζ Z})))) /@ Permutations[{a,b,x,y,z}]);

Union[AllExps,SameTest->((#1[[2]]===#2[[2]])&)]/.(p_->m_):>(p->MatrixForm[m])