Sage will compute the extended echelon form, an improvement over what we did “by hand” in Sage MISLE. And the output can be requested as a “subdivided” matrix so that we can easily work with the pieces $C$ and $L$. Pieces of subdivided matrices can be extracted with indices entirely similar to how we index the actual entries of a matrix. We will redo Example FS2 as an illustration of Theorem FS and Theorem PEEF.
Notice how we can employ the uniqueness of reduced row-echelon form (Theorem RREFU) to verify that C and L are in reduced row-echelon form. Realize too, that subdivided output is an optional behavior of the .extended_echelon_form() method and must be requested with subdivide=True.
Additionally, it is the uniquely determined matrix J that provides us with a standard version of the final column of the row-reduced augmented matrix using variables in the vector of constants, as first shown back in Example CSANS and verified here with variables defined as in Sage RRSM. (The vector method .column() converts a vector into a 1-column matrix, used here to format the matrix-vector product nicely.)