Strozzi inhabited the same palace as Lorenzino.

'When we arrived there, both Messer Ruberto and Lorenzino were leaving the house, and there were around them so many gentlemen and other persons, that I could not present myself, and both straightway stepped into the gondola.

Then I, not having seen Lorenzino for a long while past, and because he was very quietly attired, could not recognize the man exactly, but only as it were between certainty and doubt.

Wherefore I said to Spagnoletto, "I think I know that gentleman, but don't remember where I saw him." And Messer Ruberto was giving him his right hand.

Then Spagnoletto answered, "You know him well enough; he is Messer Lorenzino.
But see you tell this to nobody.