Let ℬ (D fraktur sign) be the variety of associative (special Jordan, respectively) algebras over an infinite field of characteristic 2 defined by the identity ((((x ₁, x ₂), x ₃), ((x ₄, x ₅), x ₆)), (x ₇, x ₈)) = 0 (((x ₁x ₂ · x ₃)(x ₄x ₅ · x ₆)) (x ₇x ₈ = 0, respectively). In this paper, we construct infinite independent systems of identities in the variety ℬ (D fraktur sign, respectively). This implies that the set of distinct nonfinitely based subvarieties of the variety ℬ (D fraktur sign) has the cardinality of the continuum and that there are algebras in ℬ (D fraktur sign) with undecidable word problem.