In organometallic chemistry, alpha elimination refers to reactions of this type (other spectator ligands omitted):
:X-M-CHR  →  M=CHR  +  HX
Well studied case are found in organotantalum chemistry leading to an alkylidene derivatives.  Specifically, tetraalkyl-monochloro-tantalum complex undergoes α-hydrogen elimination, followed by alkylation of the remaining chloride to give a derivative with a Ta=C bond.               
Alpha elimination contrasts with β-hydride elimination, whereby an alkyl group bonded to a metal centre is converted into the corresponding metal-bonded hydride and an alkene. 
Both α- and β-eliminations proceed via agostic intermediates.