(* Used for Definition 4.6 Standard cusp forms *) 

mA := {{1, t, t^2/2, mu t^3/6}, {0, 1, t, mu t^2/2}, {0, 0, 1, mu t}, {0, 0, 0, 1}} ;

MatrixForm[Simplify[mA]]

mAT:=MatrixForm[Collect[Simplify[mA],t]]

mAT

mATT[t_]:= (({{1, t, ((t^2)/2), ((mu (t^3))/6)}, {0, 1, t, ((mu (t^2))/2)}, {0, 0, 1, mu t}, {0, 0, 0, 1}}))

(* Check the $1$-parameter family *)

Simplify[(mATT[s].mATT[t])/.{s-> u -t}]

(*We obtain the form by differentiating and taking the translational part *)

MatrixForm[D[ (({{1, t, ((t^2)/2), ((mu (t^3))/6)}, {0, 1, t, ((mu (t^2))/2)}, {0, 0, 1, mu t}, {0, 0, 0, 1}})), t]]

(* Check invariance of 1-form. It is enough to check for the vectors. dx is invariant already. *)

Vvec1:={((mu (t^2))/2), mu t, mu}

mA3:={{1, s, s^2/2}, {0, 1, s}, {0, 0, 1}}

Simplify[mA3.Vvec1]

Vvec2:= Vvec1/.{t-> s+t}

Simplify[Vvec2]

Simplify[mA3.Vvec1 - Vvec2]