The existence domain of a fully nonlinear positive amplitude electron acoustic solitary wave has been studied in a four-component plasma composed of warm magnetized electrons, warm electron beam, and energetic multi-ion species with ions hotter than the electrons (T i > T e). A Sagdeev pseudopotential technique has been used to obtain the nonlinear evolution equation for the wave propagating obliquely with the ambient magnetic field. It is observed that the ion temperatures and concentrations play a crucial role in determining the characteristics and the existence domain of the electron acoustic solitary wave. With a large cold ion population and/or a large cold to hot ion temperature ratio, the plasma tends to behave like a single ion-dominated one. The corresponding Sagdeev pseudopotential shows an extremely narrow and deep profile producing small-amplitude, narrow width, spiky solitary waves. Such solutions are found to be applicable in the bow shock, magnetosheath, and cusp regions. Comparison with CLUSTER observations agrees well with the analytical model. It has been shown that in the magnetosheath, cooler He 2+ ions are necessary to produce a positive polarity solution while a hotter species may produce a compressive (negative polarity) solution.