The task of strapdown inertial navigation system (SINS) initial alignment is to calculate the attitude transformation matrix from body frame to navigation frame. In this paper, such attitude transformation matrix is divided into two parts through introducing the initial inertially fixed navigation frame as inertial frame. The attitude changes of the navigation frame corresponding to the defined inertial frame can be exactly calculated with known velocity and position provided by GNSS. The attitude from body frame to the defined inertial frame is estimated based on the SINS mechanization in inertial frame. The attitude, velocity and position in inertial frame are formulated together as element of the group of double direct spatial isometries.It is proven that the group state model in inertial frame satisfies a particular "group affine" property and the corresponding error model satisfies a "log linear" autonomous differential equation on the Lie algebra. Based on such striking property, the attitude from body frame to the defined inertial frame can be estimated based on the linear error model with even extreme large misalignments. Two different error state vectors are extracted based on right and left matrix multiplications and the detailed linear state space models are derived based on the right and left errors for SINS mechanization in inertial frame. With the derived linear state space models, the explicit initial alignment procedures have been presented. Extensive simulation and field tests indicate that the initial alignment based on the left error model can perform quite well within a wide range of initial attitude errors, although the used filter is still a type of linear Kalman filter. This method is promising in practical products abandoning the traditional coarse alignment stage.