  A novel invariant decomposition of diagonalizable $n \times n$ matrices into
$n$ commuting matrices is presented. This decomposition is subsequently used to
split the fundamental representation of $\mathfrak{su}(3)$ Lie algebra elements
into at most three commuting elements of $\mathfrak{u}(3)$. As a result, the
exponential of an $\mathfrak{su}(3)$ Lie algebra element can be split into
three commuting generalized Euler's formulas, or conversely, a Lie group
element can be factorized into at most three generalized Euler's formulas.
After the factorization has been performed, the logarithm follows immediately.
