We show that a primitive axial algebra of Jordan type $\eta = \tfrac{ 1}{ 2}$ is a Jordan algebra if and only if every $2$-generated subalgebra is \emph{ solid}, a notion introduced recently by Ilya Gorshkov, Sergey Shpectorov and Alexei Staroletov. As a byproduct, we show that a subalgebra generated by axes $a,b$ is solid if and only if the associator $[L_a,L_b]$ is a derivation. Moreover, we show that $2$-generated subalgebras that are not solid contain precisely $3$ axes.
Categories: math.RA math.GR