I am studying the paper, Goto, Morikuni, On algebraic Lie algebras, J. Math. Soc. Japan 1, 29-45 (1948). ZBL0038.02104. Here the author proves the unique decomposition of a matrix into a nilpotent matrix and a $s$-matrix as $X=X^0+X^s$. By a $s$-matrix he means a matrix with simple elementary divisors. He remarks that the $s$-matrix is got by taking the diagonal part of Jordan canonical form of $X$. I am not able to get how the nilpotent part is obtained. Each Jordan block is expressed as $\lambda I+N$ where $N$ is nilpotent. But how do I arrive at a nilpotent matrix? How is that $X^0$ and $X^s$ commute?
Also in the next step, he states, $X^s$ can be written as $\xi_1X_1+\xi_2X_2\ldots \xi_kX_k$ uniquely where $\xi_i$'s are taken from a suitable Hamel basis. But any fixed basis for the base field $K$ would do the job? Why Hamel basis?