Despite the insights gained in the last few years, our knowledge about the formation and evolution scenario for the spheroid-dominated galaxies is still incomplete. New and more powerful cosmological simulations have been developed that together with more precise observations open the possibility of more detailed study of the formation of early-type galaxies (ETGs). The aim of this work is to analyse the assembly histories of ETGs in a $Lambda$-CDM cosmology, focussing on the archeological approach given by the mass-growth histories.We inspected a sample of dispersion-dominated galaxies selected from the largest volume simulation of the EAGLE project. This simulation includes a variety of physical processes such as radiative cooling, star formation (SF), metal enrichment, and stellar and active galactic nucleus (AGN) feedback. The selected sample comprised 508 spheroid-dominated galaxies classified according to their dynamical properties. Their surface brightness profile, the fundamental relations, kinematic properties, and stellar-mass growth histories are estimated and analysed. The findings are confronted with recent observations.The simulated ETGs are found to globally reproduce the fundamental relations of ellipticals. All of them have an inner disc component where residual younger stellar populations (SPs) are detected. A fraction of this inner-disc correlates with bulge-to-total ratio. We find a relation between kinematics and shape that implies that dispersion-dominated galaxies with low $V/sigma_L$ (where $V$ is the average rotational velocity and $sigma_L$ the one dimensional velocity dispersion) tend to have ellipticity smaller than $sim 0.5$ and are dominated by old stars. Abridged
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