  Subspace recycling iterative methods and other subspace augmentation schemes
are a successful extension to Krylov subspace methods in which a Krylov
subspace is augmented with a fixed subspace spanned by vectors deemed to be
helpful in accelerating convergence or conveying knowledge of the solution.
Recently, a survey was published, in which a framework describing the vast
majority of such methods was proposed [Soodhalter et al, GAMM-Mitt. 2020]. In
many of these methods, the Krylov subspace is one generated by the system
matrix composed with a projector that depends on the augmentation space.
However, it is not a requirement that a projected Krylov subspace be used.
There are augmentation methods built on using Krylov subspaces generated by the
original system matrix, and these methods also fit into the general framework.
  In this note, we observe that one gains implementation benefits by
considering such augmentation methods with unprojected Krylov subspaces in the
general framework. We demonstrate this by applying the idea to the R$^3$GMRES
method proposed in [Dong et al. ETNA 2014] to obtain a simplified
implementation and to connect that algorithm to early augmentation schemes
based on flexible preconditioning [Saad. SIMAX 1997].
