Subalgebras of Lie algebra of Lorentz group will be discussed as the basic examples at the beginning of the report. The corresponding analysis is performed by canonical bases for subspaces of a vector space. All canonical bases for 5-dimensional and 4-dimensional subspaces of a 6-dimensional vector space are found, and they are utilized to find the corresponding subalgebras of 6-dimensional Lie algebra of Lorentz group.

The examples will show that each canonical basis is associated with matrix in reduced row echelon form. This relation allows us to generalize previous results, and canonical bases for (n-1)-dimensional subspaces and for (n-2)-dimensional subspaces of n-dimensional vector space will be found. Meanwhile all reduced row echelon forms of matrices and matrices will be classified also.

Prospective applications for canonical bases will be described as the final part of the report.

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