Consider a $n \times n$ triangular matrix, block-decomposed under the following form:
$$T=\begin{pmatrix}A & 0\\B & C\end{pmatrix} \ \text{where} \ A \ \text{and} \ C \ \text{are diagonal.}$$
In order to be more specific, let us consider the particular case where
$$T=\left(\begin{array}{cc|ccc}a&&&& \\ &b&&& \\ \hline p&q&f&& \\ r&s&&h&\\t&u&&&g\\\end{array}\right).$$
The general case is a straightforward generalization of this case.
Using a CAS (Mathematica), I have obtained the following formula:
$$T=\begin{pmatrix}exp(A) & 0\\D & exp(C)\end{pmatrix} \ \text{with} \ D=\left(\begin{array}{cc}p \phi(a,f)& q \phi(b,f)\\ r \phi(a,g)& s \phi(b,g)\\t \phi(a,h)& u \phi(b,h)\\\end{array}\right) \ \text{where} \ \phi(m,n):=\dfrac{e^m-e^n}{m-n}$$
I have an uninteresting proof. I would like to have indications for an insightful/heuristic proof (or reference to a paper dealing with a similar issue), that could explain the "Hadamard product" structure of block $D$ and give a rationale for the structure of function $\phi$.