On Commuting Exponentials in Low Dimensions

f, g ∈ L(E) where E is a k vector space of dimension d. We introduce the relation (*): exp(t.f + g) = exp(t.f) ◦ exp(g) for any t ∈ k or N; we study the connections between the relations () and f ◦g = g◦f for d = 2 or 3. Let d = 2: if k = R and if (*) is verified for t ∈ N then f ◦g = g◦f ; we obtain all the couples f , g verifying (*) on C and such as f ◦g 6= g◦f . Our main result is: if d = 3, k = C and exp(t.f + g) = exp(t.f) ◦ exp(g) = exp(g) ◦ exp(t.f) for t ∈ N, then f and g are simultaneously trigonalizable.