Solving the Matrix Exponential Function for Special Orthogonal Groups SO ( n ) up to n = 9 and the Exceptional Lie Group G 2

In this work the matrix exponential function is solved analytically for the special orthogonal groups S O ( n ) up to n = 9 . The number of occurring k -th matrix powers gets limited to 0 ≤ k ≤ n − 1 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 S O ( 3 ) , a quadratic equation needs to be solved for n = 4 , 5 , a cubic equation for n = 6 , 7 , and a quartic equation for n = 8 , 9 . As an interesting subgroup of S O ( 7 ) , the exceptional Lie group G 2 of dimension 14 is constructed via the matrix exponential function through a remarkably simple constraint on an invariant, ξ = 1 . The traces of the S O ( n ) -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.