  We define the notion of dextral symmetric algebras (not necessarily
associative), motivated by the idea of symmetric rings. We derive a complete
classification of dextral symmetric algebras of Leavitt path algebras, and
right Leibniz algebras up to dimension $4$. We also obtain that a
finite-dimensional dextral symmetric right Leibniz algebra is solvable if and
only if it satisfies a weaker notion of nilpotency.
