Editing Gell-Mann matrices (section)

===Trace orthonormality===

In mathematics, orthonormality typically implies a norm which has a value of unity (1).  Gell-Mann matrices, however, are normalized to a value of 2.  Thus, the [[trace (linear algebra)|trace]] of the pairwise product results in the ortho-normalization condition

:<math>\operatorname{tr}(\lambda_i \lambda_j) = 2\delta_{ij},</math>

where <math>\delta_{ij}</math> is the [[Kronecker delta]].

This is so the embedded Pauli matrices corresponding to the  three embedded subalgebras of ''SU''(2) are conventionally normalized. In this three-dimensional matrix representation,  the [[Cartan subalgebra]] is the set of linear combinations (with real coefficients) of the two matrices <math>\lambda_3</math> and <math>\lambda_8</math>, which commute with each other.

There are three [[Clebsch–Gordan_coefficients_for_SU(3)#Standard_basis|significant]] [[SU(2)]] subalgebras:

*<math>\{\lambda_1, \lambda_2, \lambda_3\}</math>
*<math>\{\lambda_4, \lambda_5, x\},</math> and
*<math>\{\lambda_6, \lambda_7, y\},</math>
where the {{mvar|x}} and {{mvar|y}} are linear combinations of <math>\lambda_3</math> and <math>\lambda_8</math>. The  SU(2) Casimirs of these subalgebras mutually commute.

However, any unitary similarity transformation of these subalgebras will yield SU(2) subalgebras. There is an uncountable number of such transformations.