The Rome "Allegory of Water" is part of a series of the elements which Jan Brueghel painted between 1606-1611. The series currently resides in the Galleria Doria Pamphilj. See also:

• Allegory of Earth (Rome)
• Allegory of Air (Rome)
• Allegory of Fire: Venus in the Forge of Vulcan (Rome)

Many copies all over the place, again; Ertz lists only "better" ones but there are also ones in Burghley House, Slot Neues Palais in Potsdam, one recently art market in Paris, Saltram House... These include a number of pretty distant variants.
Autograph variant sold Sotheby's London 8.vii.99 #53 with pendant Air; this is now Ertz #498.
Apparently independent copies (not in series) are about evenly balanced between Doria Pamphilj and Lyons versions, at least, there are a lot of each. What there are not are independent copies of the Ambrosiana version.
NB that doing elements as pendants rather than series--"Earth and Water" as one and then "Fire and Air" as another--seems to have been Jan the Younger's idea. there are lots of these, again in lots of versions--but not by Jan the Elder. There is however one version that is so much better than all the others that it seems to represent a genuine link to Jan the Elder's ideas. It isn't anywhere in Ertz. It's in the Uffizi, Photo RKD.
Ertz now accepts 2 "double element" paintings in Potsdam as Jan the Elder; I would not agree with this.
Many copies and variants in the RKD and Witt Library that are listed under the other Allegory of Water paintings.

The panel on the back show the monogram 'RB' of an unidentified Antwerp panel maker. Idem on the back of companion piece Allegory of Earth (Rome).