let be our Lie algebra. i will write for inner derivations. so we are given that for all . that means for all . call this (1). we also have, by definition of a derivation, . call this (2). now (1) and (2) gives you for all . that means is central for all and we're done.

for the second question use the fact that a matrix with distinct eigenvalues is diagonalizable. so suppose that is a basis for such that the matrix of in this basis is diagonal. that means for all . now for any define by for all where is the Kronecker delta. see that these make a basis for and

hence and so the matrix of relative to the basis is diagonal and its diagonal entries are .