In this paper the Lorentz transformation, considered as the composition of a rotation and a Lorentz boost, is decomposed into a linear combination of two orthogonal transforms. In this way a two-term expression of the Lorentz transformation by means of quaternions is proposed. An analytical solution to the problem of finding eigenvectors is given. The conditions for the existence of eigenvectors are specified. The quartet of eigenvectors, which occurs when the rotational axis is orthogonal to the velocity direction, is obtained for two cases: for the generic case of the Lorentz transformation and for the composition of the Lorentz boosts. It is shown that a quartet of eigenvectors exists for the composition of any Lorentz boosts. For the composition of boosts it is established that the half-sum (arithmetic mean) of the square roots of mutually inverse eigenvalues is found by combining half-rapidities of the original boosts according to the same cosine rule, which is used to combine the source rapidities into the resulting rapidity.