NOVELTY - Preparing 2-alkyl cephem compound (II) or its salt, involves: oxidating a compound (I) or its salt to give compound (II); or reacting a compound (IV) or its salt with base to give compound (II); or oxidating compound (I) to give a compound (V) or its salt and isomerizing compound (V) to give compound (II); or reducing a compound (XVI) or its salt to give compound (II); or reacting aldehyde, amine and a compound (XV) or its salt to give compound (XVI).    USE - For preparing 2-alkyl cephem compounds useful as antimicrobial drugs.    ADVANTAGE - The method is an efficient method for stereoselectively producing 2-alkylcephem compounds in high yield. The 2-alkyl cephem compounds or their derivatives are synthesized efficiently and conveniently, e.g. in short steps, in high yield, in high purity, in high selectivity, and/or under mild conditions. The crystals have good stability, good handling, and/or high purity.    DETAILED DESCRIPTION - Preparing a compound of formula (II) or its salt, involves: oxidating a compound of formula (I) or its salt to give the compound (II); or reacting a compound of formula (IV) or its salt, with a base to give the compound (II); or oxidating the compound (I) or its salt, to give a compound of formula (V) or its salt, and isomerizing the compound (V) to give the compound (II); or reducing a compound of formula (XVI) or its salt to give the compound (II); or reacting aldehyde, amine and a compound of formula (XV) or its salt to give the compound (XVI).    R5a=lower alkyl;    R5=H or lower alkyl;    Y'=leaving group;    L'=single bond, lower alkylene or lower alkenylene;    P1=acyl or amino-protecting group;    P2=carboxy-protecting group;    a=bond which is a mixture of alpha -configuration or beta -configuration.    INDEPENDENT CLAIMS are included for the following:    (1) preparing the compound (XVI) or its salt, involving: reacting aldehyde, amine and the compound (XV) to give the compound (XVI);    (2) preparing a compound of the formula (III) or its salt, involving: reducing the compound (II) to give the compound (III);    (3) preparing a compound of formula (Ia) or its salt, involving: reacting a compound of formula (VI) or its salt with an acid to give the compound (Ia);    (4) preparing a compound of formula (VII) or its salt, involving: deprotecting the compound (III) to give the compound (VII);    (5) new intermediate compounds (II), (III), (IV) and (VII) (excluded that L' is a single bond and Y' is halo; and L' is lower alkylene or lower alkenylene and Y' is acetyl);    (6) new intermediate compound (XVI);    (7) transforming 2- alpha -alkyl-cephem compound to 2- beta -alkyl-cephem compound, involving reacting 2- alpha -methyl-cephem compound with a base; and    (8) new crystal compounds of formulae (XVIa), (IIa) and (IIIa).    a=bond which is in cis or trans configuration.    Provided that: -L'-Y' is not -CH2-O-C(=O)-CH3.