Solving the Matrix Exponential Function for Special Orthogonal Groups SO(n) up to n = 9 and the Exceptional Lie Group G₂

In this work the matrix exponential function is solved analytically for the special orthogonal groups (Formula presented.) up to (Formula presented.). The number of occurring k-th matrix powers gets limited to (Formula presented.) by exploiting the Cayley–Hamilton relation. The corresponding expansion coefficients can be expressed as cosine and sine functions of a vector-norm V and the roots of a polynomial equation that depends on a few specific invariants. Besides the well-known case of (Formula presented.), a quadratic equation needs to be solved for (Formula presented.), a cubic equation for (Formula presented.), and a quartic equation for (Formula presented.). As an interesting subgroup of (Formula presented.), the exceptional Lie group (Formula presented.) of dimension 14 is constructed via the matrix exponential function through a remarkably simple constraint on an invariant, (Formula presented.). The traces of the (Formula presented.) -matrices arising from the exponential function are sums of cosines of several angles. This feature confirms that the employed method is equivalent to exponentiation after diagonalization, but avoids complex eigenvalues and eigenvectors and operates only with real-valued quantities.