On page 6 of this paper by Diakonov and Petrov, it is stated that the Cartan combination, $$ \sum_im_iH_i, $$ ($m_i$ are elements of a highest weight) in the fundamental representation can always be set to $$ \textrm{diag}(1,0,\ldots,0) $$ by rotating the axes and subtracting the ”unit” matrix which I understand as the identity matrix.
My questions are, how does one show this in general, and why are we allowed to subtract the identity matrix here?
For SU(2), $\textbf{m}=(1)$ in the fundamental representation and since $$H= \bigg(\begin{array} (1 \ \ \ \ \ 0 \\ 0 \ -1\end{array}\bigg), $$ subtracting the identity matrix gives me $$\bigg(\begin{array} (0 \ \ \ \ \ 0 \\ 0 \ -2\end{array}\bigg) $$
This is close to to the expected answer if we rotate the axis, and rescale by -2.
But I still do not get why the identity matrix can be subtracted?