Yes as long as the h1 b petition is filed for consular process and approved without a change of status you can continue to work on L 1 for abc as long as your current I- 94 is valid. You status will only change when you return to the U.s. on a h1 b visa
You can continue to work in L-1 status until you go abroad and return with an H-1B visa.
(213) 394-4554 x0 Mr. Shusterman is a former INS Trial Attorney (1976-82) with over 35 years of immigration experience. His response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.
A noncitizen can hold multiple visa stamps in his or her passport and use those in accordance with the purpose of a particular trip. A noncitizen can only be in one status at the time.
A noncitizen cannot change status unless the status from which the change is sought is valid on the day when the change of status application is filed.
The last action rule applies unless additional H1B employment is claimed instead of change of status. This is why it is always dangerous to file multiple petitions at the same time.
NYC EXPERIENCED IMMIGRATION ATTORNEYS www.myattorneyusa.com; email: firstname.lastname@example.org; Phone: (866) 456-8654; Fax: 212-964-0440; Cell: 212-202-0325. The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter.