How did Gaussian and Eisenstein integers get their names?

I can separate this into two questions at some point if necessary, but it's possible that sources for the answer to one will provide the answer to the other at the same time.

I learned about Eisenstein integers after studying this answer to a mathematics problem I'd asked about. Briefly they are represented by a hexagonal lattice on the complex plane, the distance of the six closest points to the origin are all unit length from it. With integers $$a$$ and $$b$$ they are

$$a + bu$$

where

$$u = \frac{1+ i \sqrt{3}}{2}.$$

Then I learned about Gaussian integers which are represented by a square lattice of length one on the complex plane. With integers $$a$$ and $$b$$ they are of the form

$$a + bi.$$

Question: Eisenstein integers are named after Gotthold Eisenstein and I assume Gaussian integers are named after Carl Friedrich Gauss, but who gave these names to these number sets in the complex plane?

Or at least how did consensuses for their names arise?

• Jeff Miller's website catalogs this sort of thing:"G.H. Hardy and E.M. Wright refer to the ‘complex’ or ‘Gaussian’ integers"... (An Introduction to the Theory of Numbers, 1938)". There is nothing on Eisenstein integers' naming. – Conifold Dec 1 '20 at 12:52
• @Conifold that's quite a resource, thank you! – uhoh Dec 1 '20 at 13:22