You are right. Both Oscillatoria and Lyngbya are two genera of Blue green algae (B.G) which are classified as Cyanobacteria (term introduced by Stanier ex Cavalier-Smith 2002.)(Cyanoprokaryotes) - representing one of the most ancient groups of organisms on earth, with some fossil representatives having morphology very similar to present-day species (Schopf 1974,Knoll 2008).
I would like to call them "the pioneer oxygenizers of the planet". Fritsch FE (1945) classified them under the class Myxophyceae (Cyanophyceae-by Schaffner, 1909) but mentioned that it is also called Schizophyceae or Phycochromophyceae.The Phycologists (after Fritsch) like Desikachary ,Prescott ,Morris,Round etc..included the B.G under the division Cyanophyta though Chapman (1962) put them under the division Myxophycophyta. Bold and Wynne (1978) placed them under the division Cyanochloronta. Parker (1982) used Cyanophycota under the Group I Prokaryota.
As per the modern systems, Cyanobacteria are classified on the basis of multiple criteria (Hoffmann et al. 2005) into six orders (Chroococcales, Gloeobaterales, Nostocales, Oscillatoriales, Pseudanabaenales, and Synechococcales.---Thomazeae et al. (2011)
Systematic position of Oscillatoria and Lyngbya:
As per the current classification as appear on the website "Algae base "they are two genera under the family Oscillatoriaceae.
Both of them belong to the same family as Dr. Jackson has clearly indicated. But I am little bit confused with your following statement "Taxonomy of algae must be clearly understood". What you really meant by this?? Would you please clarify it!!!
Yes, you are right, both are Cyanobacteria. Pay attention with the genus Lyngbya: the molecular studies showed that it should be separated in different genera. See the paper:
Niclas Engene, R. Cameron Coates, and William H. Gerwick. 2010. 16S rRNA GENE HETEROGENEITY IN THE FILAMENTOUS MARINE
CYANOBACTERIAL GENUS LYNGBYA. Journal of Phycology 46