Why is the lancelet a chordate, but not a vertebrate?

Lancelet is a chordate but not a vertebrate because it lacks a backbone

Explanation:

The subphyla of the phylum Chordata are - Vertebrata, Urochordata and cephalochordate

Those chordates with a structural backbone belong to Vertebrata.

Urochordates and cephalochordates lack a backbone and hence are termed invertebrate chordates.

Lancelet is an invertebrate chordate of the subphylum – Cephalochordata.  

The lancelets consist of a notochord that extends right from the head till the tail at the dorsal surface. The notochord acts like the backbone and provides support and a semi-flexible structure to aid in the movement of the organism.

They lack a developed nervous system like vertebrates but only has a dorsal nerve tube .

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voluntary muscles are controlled by the cerebellum.