In the continuity of our previous paper arXiv:1509.05516, we define three new algebras, $A_{ n}(a,b,c)$, $B_{ n}$ and $C_{ n}$, that are close to the braid algebra. They allow to build solutions to the braided Yang-Baxter equation with spectral parameters. The construction is based on a baxterisation procedure, similar to the one used in the context of Hecke or BMW algebras. The $A_{ n}(a,b,c)$ algebra depends on three arbitrary parameters, and when the parameter $a$ is set to zero, we recover the algebra $M_{ n}(b,c)$ already introduced elsewhere for purpose of baxterisation. The Hecke algebra (and its baxterisation) can be recovered from a coset of the $A_{ n}(0,0,c)$ algebra. The algebra $A_{ n}(0,b,-b^2)$ is a coset of the braid algebra. The two other algebras $B_{ n}$ and $C_{ n}$ do not possess any parameter, and can be also viewed as a coset of the braid algebra.
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