I have a question when I study the Lie algebra.
The following sentence is written in the book related to Lie algebra, but the contents is just a linear algebra:
Since $diag(a,\cdots,a)$ commutes with the nilpotent matrix having one's just above the diagonal and zeros elsewhere, $x$ is the sum of a diagonal and a nilpotent matrix which commute. We can make this decomposition more precise, as follows.
This book is "Introduction to Lie Algebras and Representation theory, James E. Humphreys, Second printing, revised", and the page is 17.
I cannot understand why the contition "commute" is the crucial reason for the expression of the sum of a diagonal and a nilpotent matrix. What is the connection between them? I'm wating your best answer. Thanks.
Best.