Cocalibrated G*2-structures and cocalibrated -structures are the natural initial values for Hitchin's evolution equations whose solutions define (pseudo)-Riemannian manifolds with holonomy group contained in Spin(7) or Spin0(3, 4), respectively. In this article, we classify 7-D real Lie algebras with a codimension one Abelian ideal which admit such structures. Moreover, we classify the 7-D complex Lie algebras with a codimension one Abelian ideal which admit cocalibrated G2C-structures.