The talk is going to be on progress made in approximation theory of Lie group-valued periodic functions (loops) by so-called polynomial loops. This is a relatively unexplored topic within the larger area of nonlinearly constrained approximation, which includes the study of H\"{o}lder classes of Lie-group-valued functions, smoothness preserving factorization techniques, but also interfaces with splitting methods for the exponential operators and practical applications (e.g., orthogonal wavelet construction). The technical part is based on application of higher order splitting formulas to the exponential map exp(t(A_1+...+A_n)), with A_i belonging to the corresponding Lie algebra, which leads to deriving "nearly optimal" asymptotic rates in approximation.