The two compact real form Lie algebras $\mathfrak{so}(16)$ and $\mathfrak{su}(11)$ have the same dimension (120).
They are certainly not isomorphic, but does there exist some kind of algebraic procedure (applying to the generators of these Lie algebras) to go from one to the other?
I do know of one strange thing that is kind of like this!
There are two real Clifford algebras corresponding to two different signature choices (+--- and -+++). Both of these signature conventions can be used to do things with Minkowski spacetime.
They are the same dimension, and have complementary signatures... but the two Clifford algebras are nonisomorphic!
However Pertti Lounesto shows in Clifford Algebras and Spinors that there is a "tilting map" that connects the two algebras. I thought I remember scanning it in detail once, but now I can't seem to access that page on googlebooks.