Key having a retractable insert with limited clearance

A key, for a motor vehicle, comprising:

• a casing, with an upper half-shell forming a cover and a lower half-shell forming a bottom of the casing;
• an insert, comprising a bit and a bit support, mounted pivotably relative to said casing between a rest position in which the insert is retracted into said casing and a position of use in which the insert is deployed relative to said casing;
• a mechanism for deploying the insert mounted in said casing comprising a push button;
• an upper rim of an upper movement area, incorporated in an inner surface of the upper half-shell, of the support, and a lower rim of a lower movement area, incorporated in an inner surface of the lower half-shell, of the support, the upper rim and the lower rim extending between a key rotation end and a blocking end, the upper movement area and the lower movement area forming a movement sector of the support;
• a first play between the rotation end and the support;
• characterized in that the movement sector of the support comprises at least one constriction point, exhibiting a second play, smaller than the first play, in relation to said support.