A pair of Lie algebras \((L,L_0)\) is called transitive if \(L_0\) does not contain any non-trivial ideal of \(L\). Let \(L_1=\{x\in L_0\mid [x,L]\subset L_0\}\). Then \(L_0\) is called an ample nonlinear subalgebra of \(L\) if \(L_1\neq\{0\}\) and \(L_0=N_L(L_1)\) (normalizer), and \(L_1\) is called the kernel of \(L_0\). The main results in chapters 7 and 8 of this paper consist of a series of classification results over the real and complex numbers for ample nonlinear subalgebras and their kernels. In particular, the ample nonlinear subalgebras in a semisimple Lie algebra \(L\) are precisely the ample parabolic subalgebras of \(L\).