  Of four types of Kaplansky algebras, type-2 and type-4 algebras have
previously unobserved $\mathbb{Z}/2$-gradings: nonlinear in roots. A method
assigning a simple Lie superalgebra to every $\mathbb{Z}/2$-graded simple Lie
algebra in characteristic 2 is illustrated by seven new series. Type-2 algebras
and one of the two type-4 algebras are demystified as nontrivial deforms (the
results of deformations) of the alternate Hamiltonian algebras. The type-1
Kaplansky algebra is recognized as the derived of the nonalternate version of
the Hamiltonian Lie algebra, the one that preserves a tensorial 2-form, not an
exterior one.
  Deforms corresponding to nontrivial cohomology classes can be isomorphic to
the initial algebra, e.g., we confirm Grishkov's implicit claim and explicitly
describe the Jurman algebra as such a "semitrivial" deform of the derived of
the alternate Hamiltonian Lie algebra. This paper helps to sharpen the
formulation of a conjecture describing all simple finite-dimensional Lie
algebras over any algebraically closed field of nonzero characteristic and
supports a conjecture of Dzhumadildaev and Kostrikin stating that all simple
finite-dimensional modular Lie algebras are either of "standard" type or
deforms thereof.
  In characteristic 2, we give sufficient conditions for the known deformations
to be semitrivial.
