Mohs hardness measures resistance to scratching, not resistance to every kind of damage. Lapis is softer at 5 to 6. Aquamarine is 7.5 to 8, and topaz is 8, but topaz has cleavage and can still be damaged by a hard blow. In a bracelet, exposed bead edges, stringing, metal plating, and contact with watches or countertops also matter.
Use warm soapy water conservatively and follow the most delicate component in a mixed piece. Avoid steam and ultrasonic cleaning for lapis and for assembled jewelry that includes mother-of-pearl or plated metal.