I would like to compute $N_{mathfrak{gl}(n)}(mathfrak{d_n})$, $N_{mathfrak{gl}(n)}(mathfrak{u_n})$ and $N_{mathfrak{gl}(n)}(mathfrak{l_n})$, where $mathfrak{d_n}, mathfrak{u_n}, mathfrak{l_n}$ are the Lie algebras of diagonal matrices, upper triangular matrices and lower triangular matrices respectively.

By definition, $N_{mathfrak{gl}(n)}(mathfrak{d_n})= left{ x in mathfrak{gl}(n) : (x, D) in mathfrak{d_n}, forall D in mathfrak{d_n} right}$. By linearity, every $x in mathfrak{gl_n}$ is a combination of elementary matrices of type $E_{ij}$, so it will be enough to examine $(E_{ij}, E_{kk}) = delta_{jk}E_{ik} – delta_{ki}E_{kj}$ which is a diagonal matrix every time $j=k=i$. So, I conclude that $N_{mathfrak{gl}(n)}(mathfrak{d_n}) = mathfrak{d_n}$. Is this correct?

As for $N_{mathfrak{gl}(n)}(mathfrak{u_n})$, one can first notice that $x in mathfrak{gl}(n) iff x= u +l$ with $u in mathfrak{u_n}$ and $l in mathfrak{l_n}$. So, $t in mathfrak{u_n}$, $(x,t) = (u,t) + (l,t)$. Obviously $(u, t) in mathfrak{u_n}$ but I am not sure what conditions are to be made on $l$ such that $(l,t) in mathfrak{u_n}$ as well. I guess my question is: when is the bracket of an upper triangular and lower triangular matrix equal to an upper triangular?