What is the legal test for a defendant pleading not guilty to a lesser included crime?

"When a defendant pleads not guilty, the court lacks jurisdiction to convict him of an offense that is neither charged nor necessarily included in the alleged crime. [Citations.] This reasoning rests upon a constitutional basis: 'Due process of law requires that an accused be advised of the charges against him in order that he may have a reasonable opportunity to prepare and present his defense and not be taken by surprise by evidence offered at his trial.' [Citation.] ..." (People v. West (1970) 3 Cal.3d 595, 612 [91 Cal.Rptr. 385, 477 P.2d 409].)

"As to a lesser included offense, the required notice is given when the specific language of the accusatory pleading adequately warns the defendant that the People will seek to prove the elements of the lesser offense. [Citations.] However, even when the charge does not so specify, the requisite notice is nonetheless afforded if the lesser offense is 'necessarily included' within the statutory definition of the charged offense; in such event conviction of the included offense is expressly authorized ( 1159)...." (People v. Lohbauer (1981) 29 Cal.3d 364, 368-369 [173 Cal.Rptr. 453, 627 P.2d 183.)