Yes. An L-2 spouse is employment-authorized incident to status, which means the authorization comes with the classification rather than from a separate application, and the work is not limited to the principal's employer. Children in L-2 status may attend school but are not authorized to work.
The spouse's rights follow the classification, not a separate approval
This is one of the sharper practical differences between categories, and it often decides which route a household prefers. Because the authorization is incident to status, an L-2 spouse may take a job with any employer, including a competitor of the transferring company, and may start a business. Documentation practice still matters: employers completing employment eligibility verification will want to see the admission record, and it is sensible to carry the marriage certificate and the principal's approval notice.
Two limits deserve attention in Halina's case. Her spouse's authorization ends when her own status does, so any lapse on her side ends his employment as well. And a child in L-2 status can study at any level but cannot work, so a nineteen-year-old planning a summer job at the elevator would need a different basis for it.