# linear algebra – Computing normaliser of diagonal and triangular matrices

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?