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In field characteristic $p$, the identity matrix has trace $p=0$, and hence belongs to ${\mathfrak{sl}}_p$. Its span is a nonzero proper central ideal, so ${\mathfrak{sl}}_p$ is not a simple Lie algebra. In fact its center of a Lie algebra is precisely $k{I}_p$: commuting with every off-diagonal matrix unit forces a matrix to be scalar.

For $p>2$, the matrix-unit extraction lemma for special linear ideals still works because two is invertible. Any ideal containing a nonscalar matrix contains every off-diagonal matrix unit and every trace-zero diagonal matrix, hence equals ${\mathfrak{sl}}_p$. Therefore an ideal of the quotient Lie algebra pulls back either to the center or to the whole algebra. The quotient is nonabelian, since $[E_{12},E_{23}]=E_{13}$ remains nonzero modulo the center. ConsequentlyNo separation of diagonal root spaces is needed, so this proof also handles $p=3$ without assuming their weights are distinct.

For $p=2$, every trace-zero two-by-two matrix has the form $aI+b{E}_{12}+c{E}_{21}$. The only nonzero bracket between these basis vectors is $[E_{12},E_{21}]=I$, which is central. Thus