Solid lines in axial algebras of Jordan type \tfrac{1}2 and Jordan algebras

We show that a primitive axial algebra of Jordan type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="one half"> <mml:semantics> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> <mml:annotation encoding="application/x-tex">\tfrac {1}{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Jordan algebra if and only if every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -generated subalgebra is <italic>solid</italic> , a notion introduced recently by Ilya Gorshkov, Sergey Shpectorov and Alexei Staroletov. As a byproduct, we show that a subalgebra generated by axes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a comma b"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">a,b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is solid if and only if the associator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper L Subscript a Baseline comma upper L Subscript b Baseline right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>a</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[L_a,L_b]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a derivation. Moreover, we show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -generated subalgebras that are not solid contain precisely <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> axes.