Clear[Ahat]
Ahat = {{a[w], 
              b[w], a'[w]*u + b'[w]*v}, {c[w], 
              d[w], c'[w]*u + d'[w]*v}, {0, 0, f'[w]}}
MatrixForm[Ahat]
{EV1, EV2, EV3} = Eigenvectors[Ahat];
{LEV1, LEV2, LEV3} = 
            Inverse[Transpose[{EV1, EV2, EV3}]];
MatrixForm[{LEV1, LEV2, LEV3}]
Dop[
            f_] := {(∂_u) f, (∂_v) f, (∂_w) 
              f}
LB[V1_, V2_] := 
          Dop[V1] . V2 - Dop[V2] . V1

"The Lie Brackets are"
MatrixForm[Simplify[LB[EV1, EV2]]]
MatrixForm[Simplify[LB[EV2, EV3]]]
MatrixForm[Simplify[LB[EV1, EV3]]]

"Here is a way to find the riemann invarients of LEV1,2"
MatrixForm[Simplify[LEV1 ]/(c[w] (-(((a[w] - 
                                d[w] + (Sqrt[a[w]^2 + 4 b[w] c[w] - 2 a[w] d[w] + d[w]^2])) (c[w] (u a'[w]+v b'[w])-(u c'[w]+v d'[w]) (a[w]-f'[w])))/(c[
                            w]))+2 (-d[w] (u a'[w]+v b'[w])+b[w] (u c'[w]+v d'[w])+(u a'[w]+v b'[w]) f'[w])))/(2 (Sqrt[a[w]^2 + 4 b[w] c[w] - 2 a[w] d[w] + 
                      d[w]^2]) (b[w] c[w]+(a[w]-f'[w]) (-d[w]+f'[w])))]