Let $M \in \mathfrak{u}(n)$ be some hermitian matrix, and let $\{t_a\}, a=1,...,n^2$, be some basis for $\mathfrak{u}(n)$. Since $M$ is diagonalizable, we can write $M = UDU^\dagger$, where $D$ is the matrix of eigenvalues of $M$ and $U \in U(n)$ is some unitary matrix that diagonalizes $M$. Now, $D \in \mathfrak{u}(n)$, so we can expand $D = d^ih_i$, where $d^i$ are some expansion coefficients (in fact, $d^i = Tr(Dh^i)$) and $\{h_i\}, i = 1,..., n$, is a subset of $\{t_a\}$ consisting only of Cartan generators (the diagonal generators). The diagonalization is now
$M = d^iUh_iU^\dagger = d^iAd_Uh_i = d^i\Lambda_{ai}(U)t_a$, (1)
where $Ad_Uh_i = Uh_iU^\dagger$ is the adjoint representation acting on Cartan generators and $\Lambda_{ai}$ is the matrix representation of this linear transformation. Now, on the left hand side of equation (1), we have $n^2$ coordinates, $M_{ij}$, and on the right hand side we have $n + n^3$ ones, $(d^i, \Lambda_{ai})$. To be consistent, we must have some constraints that reduces the number of independent components of $\Lambda$ to $n(n-1)$. So, it seems to me that the expansion in the right hand side of (1) will have only $n-1$ generators $t_a$, which is really weird to me, but I cannot find my mistake. What's wrong with those considerations?