= Matrix-unit extraction lemma for special linear ideals
Over a <field> of <characteristic> different from two, any <Lie algebra ideal> in $\mathfrak{sl}_n$ containing a nonscalar <matrix> is the whole <special linear Lie algebra>. Indeed, if $A_{ji}\ne0$ for $i\ne j$, then $[E_{ij},[E_{ij},A]]=-2A_{ji}E_{ij}$ extracts a <matrix unit>. For a nonscalar diagonal <matrix>, $[E_{ij},A]=(A_{jj}-A_{ii})E_{ij}$ extracts one instead. Its bracket with $E_{ji}$ produces $H=E_{ii}-E_{jj}$, and $[H,E_{ji}]=-2E_{ji}$ extracts the reverse unit. Brackets $[E_{ij},E_{jk}]=E_{ik}$ and $[E_{ki},E_{ij}]=E_{kj}$ spread these units to every row and column; opposite pairs span all trace-zero diagonal <matrices>. Thus only ideals consisting of scalar <matrices> can be proper. This proof works even when restrictions of distinct <roots of a root system> to a <Cartan subalgebra> coincide in positive <characteristic>.
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