Let \(G\) be a connected real Lie group with Lie algebra \(L(G)\), and let \(\exp:L (G)\to G\) be the exponential map of \(G\). It is an interesting problem to determine the image of the exponential map. An element of \(G\) is called exponential if it belongs to the image \(\exp(L(G))\), and \(G\) is called exponential if \(G=\exp (L(G))\). The situation is reasonably well understood for solvable Lie groups, partial results are known for semisimple Lie groups, but little is known for general Lie groups [cf. D. Z. Djoković and K. H. Hofmann, J. Lie Theory 7, 171-199 (1997; Zbl 0888.22003)]. This paper treats the exponentiality question for a semidirect product connected Lie group \(G=K\cdot R\), where \(K\) is a compact subgroup and \(R\) is a simply connected exponential solvable normal subgroup of \(G\). The main result is as follows. Let \(N\) be the nilradical of \(R\). If for every \(x\in N\) the subgroup \(\{k\in K\mid kx= xk\}\) is connected, then \(G\) is exponential. The authors apply this result to study the exponentiality for matrix groups with quaternionic entries. For \(n\geq 2\) let \(Q_n\) be the group of \(n\times n\) invertible upper triangular matrices with quaternionic entries, \(M_n\) be the subgroup of \(Q_n\) consisting of all elements whose diagonal entries are all of quaternionic norm 1, and let \(G\) be a connected Lie subgroup of \(Q_n\) containing \(M_n\). Then, \(G\) is exponential if \(n\leq 4\) and \(G\) is not exponential if \(n\geq 8\).