# Name of upper triangular/lower triangular Lie Algebra decomposition

What is the name of the Lie algebra decomposition where the positive root vectors are upper triangular and the negative root vectors are lower triangular?

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What do you mean by upper triangular and lower triangular? Do you have a specific matrix realization of your (semisimple?) Lie algebra? – Steven Sam Jan 29 '10 at 10:42
Good point, yes. Consider the fundamental representation of the semisimple Lie algebra and then conjugate by the general Linear group to obtain the form mentioned in the question for the representation matrices. – Q.Q.J. Jan 29 '10 at 11:56