 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 $A_{ji}\ne 0$ 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.