Projective symplectic group is simple

Statement
Suppose $$m$$ is a positive integer and $$k$$ is a field. Then, the projective symplectic group $$PSp(2m,k)$$ is a simple group except in the following three cases: $$m = 1$$ and $$k$$ has two elements, $$m = 1$$ and $$k$$ has three elements, $$m = 2$$ and $$k$$ has two elements. In other words, the three exceptions are $$PSp(2,2),PSp(2,3),PSp(4,2)$$.

Related facts

 * Symplectic group is perfect
 * Symplectic group is quasisimple
 * Projective special linear group is simple
 * Special linear group is perfect
 * Special linear group is quasisimple