(1)

\begin{align*} \left\{ A,B\right\} & =AB+BA\\ & =BA+AB\\ & =\left\{ B,A\right\} \end{align*}

(2)

\begin{align*} \left\{ A,B+C\right\} & =A\left(B+C\right)+\left(B+C\right)A\\ & =AB+BA+AC+CA\\ & =\left\{ A,B\right\} +\left\{ A,C\right\} \end{align*}

(3)

\begin{align*} \left\{ AB,C\right\} & =\left[AB,C\right]+2CAB\\ & =A\left[B,C\right]+\left[A,C\right]B+2CAB\\ & =A\left[B,C\right]+\left\{ A,C\right\} B-2CAB+2CAB\\ & =A\left[B,C\right]+\left\{ A,C\right\} B\\ & =\left\{ A,C\right\} B+A\left[B,C\right] \end{align*} \begin{align*} \left\{ AB,C\right\} & =\left[AB,C\right]+2CAB\\ & =A\left[B,C\right]+\left[A,C\right]B+2CAB\\ & =A\left\{ B,C\right\} -2ACB+\left[A,C\right]B+2CAB\\ & =A\left\{ B,C\right\} -2\left[A,C\right]B+\left[A,C\right]B\\ & =A\left\{ B,C\right\} -\left[A,C\right]B \end{align*}

(4)

\begin{align*} \left\{ A,BC\right\} & =\left\{ BC,A\right\} \\ & =B\left\{ C,A\right\} -\left[B,A\right]C\cmt{\because\left\{ AB,C\right\} =A\left\{ B,C\right\} -\left[A,C\right]B}\\ & =B\left\{ A,C\right\} +\left[A,B\right]C \end{align*} \begin{align*} \left\{ A,BC\right\} & =\left\{ BC,A\right\} \\ & =\left\{ B,A\right\} C+B\left[C,A\right]\cmt{\because\left\{ AB,C\right\} =\left\{ A,C\right\} B+A\left[B,C\right]}\\ & =\left\{ A,B\right\} C-B\left[A,C\right] \end{align*}

(5)

\begin{align*} \left\{ A_{1}A_{2}\cdots A_{m},B\right\} & =\left\{ A_{1}\cdots A_{p},B\right\} A_{p+1}\cdots A_{m}+A_{1}\cdots A_{p}\left[A_{p+1}\cdots A_{m},B\right]\cmt{\because\left\{ AB,C\right\} =\left\{ A,C\right\} B+A\left[B,C\right]}\\ & =\left(A_{1}\cdots A_{p-1}\left\{ A_{p},B\right\} -\left[A_{1}\cdots A_{p-1},B\right]A_{p}\right)A_{p+1}\cdots A_{m}+A_{1}\cdots A_{p}\left[A_{p+1}\cdots A_{m},B\right]\cmt{\because\left\{ AB,C\right\} =A\left\{ B,C\right\} -\left[A,C\right]B}\\ & =A_{1}\cdots A_{p-1}\left\{ A_{p},B\right\} A_{p+1}\cdots A_{m}-\left[A_{1}\cdots A_{p-1},B\right]A_{p}A_{p+1}\cdots A_{m}+A_{1}\cdots A_{p}\left[A_{p+1}\cdots A_{m},B\right]\\ & =A_{1}\cdots A_{p-1}\left\{ A_{p},B\right\} A_{p+1}\cdots A_{m}-\sum_{j\in\left\{ 1,2,\cdots p-1\right\} }A_{1}\cdots A_{j-1}\left[A_{j},B\right]A_{j+1}\cdots A_{p-1}A_{p}A_{p+1}\cdots A_{m}+\sum_{j\in\left\{ p+1,\cdots,m\right\} }A_{1}\cdots A_{p}A_{p+1}\cdots A_{j-1}\left[A_{j},B\right]A_{j+1}\cdots A_{m}\\ & =A_{1}\cdots A_{p-1}\left\{ A_{p},B\right\} A_{p+1}\cdots A_{m}-\sum_{j\in\left\{ 1,2,\cdots p-1\right\} }A_{1}\cdots A_{j-1}\left[A_{j},B\right]A_{j+1}\cdots A_{m}+\sum_{j\in\left\{ p+1,\cdots,m\right\} }A_{1}\cdots A_{j-1}\left[A_{j},B\right]A_{j+1}\cdots A_{m}\\ & =\sum_{j\in\left\{ 1,2,\cdots,m\right\} }A_{1}\cdots A_{j-1}\left(\delta_{j,p}\left\{ A_{j},B\right\} +\sgn\left(j-p\right)\left[A_{j},B\right]\right)A_{j+1}\cdots A_{m} \end{align*} \(p=1\)のときは
\begin{align*} \left\{ A_{1}A_{2}\cdots A_{m},B\right\} & =\sum_{j\in\left\{ 1,2,\cdots,m\right\} }A_{1}\cdots A_{j-1}\left(\delta_{j,p}\left\{ A_{j},B\right\} +\sgn\left(j-p\right)\left[A_{j},B\right]\right)A_{j+1}\cdots A_{m}\\ & =\sum_{j\in\left\{ 1,2,\cdots,m\right\} }A_{1}\cdots A_{j-1}\left(\delta_{j,1}\left\{ A_{j},B\right\} +\sgn\left(j-1\right)\left[A_{j},B\right]\right)A_{j+1}\cdots A_{m}\\ & =\left(\delta_{1,1}\left\{ A_{1},B\right\} +\sgn\left(1-1\right)\left[A_{1},B\right]\right)A_{1+1}\cdots A_{m}+\sum_{j\in\left\{ 2,3,\cdots,m\right\} }A_{1}\cdots A_{j-1}\left(\delta_{j,1}\left\{ A_{j},B\right\} +\sgn\left(j-1\right)\left[A_{j},B\right]\right)A_{j+1}\cdots A_{m}\\ & =\left\{ A_{1},B\right\} A_{2}\cdots A_{m}+\sum_{j\in\left\{ 2,3,\cdots,m\right\} }A_{1}\cdots A_{j-1}\left[A_{j},B\right]A_{j+1}\cdots A_{m} \end{align*} となる。
または直接計算すると、
\begin{align*} \left\{ A_{1}A_{2}\cdots A_{m},B\right\} & =\left\{ A_{1},B\right\} A_{2}\cdots A_{m}+A_{1}\left[A_{2}\cdots A_{m},B\right]\cmt{\because\left\{ AB,C\right\} =\left\{ A,C\right\} B+A\left[B,C\right]}\\ & =\left\{ A_{1},B\right\} A_{2}\cdots A_{m}+A_{1}\sum_{j\in\left\{ 2,3\cdots,m\right\} }A_{2}\cdots A_{j-1}\left[A_{j},B\right]A_{j+1}\cdots A_{m}\\ & =\left\{ A_{1},B\right\} A_{2}\cdots A_{m}+\sum_{j\in\left\{ 2,3\cdots,m\right\} }A_{1}A_{2}\cdots A_{j-1}\left[A_{j},B\right]A_{j+1}\cdots A_{m} \end{align*} となる。
\(p=m\)のときは、
\begin{align*} \left\{ A_{1}A_{2}\cdots A_{m},B\right\} & =A_{1}\left\{ A_{2}A_{3}\cdots A_{m},B\right\} -\left[A_{1},B\right]A_{2}\cdots A_{m}\\ & =A_{1}\left(A_{2}\left\{ A_{3}\cdots A_{m},B\right\} -\left[A_{2},B\right]A_{3}\cdots A_{m}\right)-\left[A_{1},B\right]A_{2}\cdots A_{m}\\ & =A_{1}A_{2}\left\{ A_{3}\cdots A_{m},B\right\} -A_{1}\left[A_{2},B\right]A_{3}\cdots A_{m}-\left[A_{1},B\right]A_{2}\cdots A_{m}\\ & =\cdots\\ & =A_{1}A_{2}\cdots A_{m-1}\left\{ A_{m},B\right\} -\sum_{j\in\left\{ 1,2,\cdots,m-1\right\} }A_{1}\cdots A_{j-1}\left[A_{j},B\right]A_{j+1}\cdots A_{m} \end{align*} となる。

(6)

\begin{align*} \left\{ A,B_{1}B_{2}\cdots B_{n}\right\} & =\left\{ B_{1}B_{2}\cdots B_{n},A\right\} \\ & =\sum_{j\in\left\{ 1,2,\cdots,m\right\} }B_{1}\cdots B_{j-1}\left(\delta_{j,q}\left\{ B_{j},A\right\} +\sgn\left(j-q\right)\left[B_{j},A\right]\right)B_{j+1}\cdots B_{n}\cmt{\because\left\{ A_{1}A_{2}\cdots A_{m},B\right\} =\sum_{j\in\left\{ 1,2,\cdots,m\right\} }A_{1}\cdots A_{j-1}\left(\delta_{j,p}\left\{ A_{j},B\right\} +\sgn\left(j-p\right)\left[A_{j},B\right]\right)A_{j+1}\cdots A_{m}}\\ & =\sum_{j\in\left\{ 1,2,\cdots,m\right\} }B_{1}\cdots B_{j-1}\left(\delta_{j,q}\left\{ A,B_{j}\right\} -\sgn\left(j-q\right)\left[A,B_{j}\right]\right)B_{j+1}\cdots B_{n} \end{align*} となる。
\(q=1\)のときは直接計算すると、
\begin{align*} \left\{ A,B_{1}B_{2}\cdots B_{n}\right\} & =\left\{ A,B_{1}\right\} B_{2}\cdots B_{n}-B_{1}\left[A,B_{2}\cdots B_{n}\right]\cmt{\because\left\{ A,BC\right\} =\left\{ A,B\right\} C-B\left\{ A,C\right\} }\\ & =\left\{ A,B_{1}\right\} B_{2}\cdots B_{n}-B_{1}\sum_{j\in\left\{ 2,3,\cdots,n\right\} }B_{2}\cdots B_{j-1}\left[A,B_{j}\right]B_{j+1}\cdots B_{n}\\ & =\left\{ A,B_{1}\right\} B_{2}\cdots B_{n}-\sum_{j\in\left\{ 2,3,\cdots,n\right\} }B_{1}B_{2}\cdots B_{j-1}\left[A,B_{j}\right]B_{j+1}\cdots B_{n} \end{align*} となり、\(q=n\)のときは直接計算すると、
\begin{align*} \left\{ A,B_{1}B_{2}\cdots B_{n}\right\} & =B_{1}\cdots B_{n-1}\left\{ A,B_{n}\right\} +\left[A,B_{1}\cdots B_{n-1}\right]B_{n}\cmt{\because\left\{ A,BC\right\} =B\left\{ A,C\right\} +\left[A,B\right]C}\\ & =B_{1}\cdots B_{n-1}\left\{ A,B_{n}\right\} +\left(\sum_{j\in\left\{ 1,2,\cdots,n-1\right\} }B_{1}\cdots B_{j-1}\left[A,B_{j}\right]B_{j+1}\cdots B_{n-1}\right)B_{n}\\ & =\left\{ A,B_{1}\right\} B_{2}\cdots B_{n}-\sum_{j\in\left\{ 2,3,\cdots,n-1\right\} }B_{1}\cdots B_{j-1}\left[A,B_{j}\right]B_{j+1}\cdots B_{n} \end{align*} となる。

(7)

\begin{align*} \left\{ AB,CD\right\} & =\left\{ A,CD\right\} B+A\left[B,CD\right]\cmt{\because\left\{ AB,C\right\} =\left\{ A,C\right\} B+A\left[B,C\right]}\\ & =\left(C\left\{ A,D\right\} +\left[A,C\right]D\right)B+A\left(\left[B,C\right]D+C\left[B,D\right]\right)\cmt{\because\left\{ A,BC\right\} =B\left\{ A,C\right\} +\left[A,B\right]C}\\ & =C\left\{ A,D\right\} B+\left[A,C\right]DB+A\left[B,C\right]D+AC\left[B,D\right] \end{align*} \begin{align*} \left\{ AB,CD\right\} & =\left\{ A,CD\right\} B+A\left[B,CD\right]\cmt{\because\left\{ AB,C\right\} =\left\{ A,C\right\} B+A\left[B,C\right]}\\ & =\left(\left\{ A,C\right\} D-C\left[A,D\right]\right)B+A\left(\left[B,C\right]D+C\left[B,D\right]\right)\cmt{\because\left\{ A,BC\right\} =\left\{ A,B\right\} C-B\left[A,C\right]}\\ & =\left\{ A,C\right\} DB-C\left[A,D\right]B+A\left[B,C\right]D+AC\left[B,D\right] \end{align*} \begin{align*} \left\{ AB,CD\right\} & =A\left\{ B,CD\right\} -\left[A,CD\right]B\cmt{\because\left\{ AB,C\right\} =A\left\{ B,C\right\} -\left[A,C\right]B}\\ & =A\left(C\left\{ B,D\right\} +\left[B,C\right]D\right)-\left(\left[A,C\right]D+C\left[A,D\right]\right)B\cmt{\because\left\{ A,BC\right\} =B\left\{ A,C\right\} +\left[A,B\right]C}\\ & =AC\left\{ B,D\right\} +A\left[B,C\right]D-\left[A,C\right]DB-C\left[A,D\right]B \end{align*} \begin{align*} \left\{ AB,CD\right\} & =A\left\{ B,CD\right\} -\left[A,CD\right]B\cmt{\because\left\{ AB,C\right\} =A\left\{ B,C\right\} -\left[A,C\right]B}\\ & =A\left(\left\{ B,C\right\} D-C\left[B,D\right]\right)-\left(\left[A,C\right]C+C\left[A,D\right]\right)B\cmt{\because\left\{ A,BC\right\} =\left\{ A,B\right\} C-B\left[A,C\right]}\\ & =A\left\{ B,C\right\} D-AC\left[B,D\right]-\left[A,C\right]DB-C\left[A,D\right]B \end{align*} \begin{align*} \left\{ AB,CD\right\} & =C\left\{ AB,D\right\} +\left[AB,C\right]D\cmt{\because\left\{ A,BC\right\} =B\left\{ A,C\right\} +\left[A,B\right]C}\\ & =C\left(\left\{ A,D\right\} B+A\left[B,D\right]\right)+\left(\left[A,C\right]B+A\left[B,C\right]\right)D\cmt{\because\left\{ AB,C\right\} =\left\{ A,C\right\} B+A\left[B,C\right]}\\ & =C\left\{ A,D\right\} B+CA\left[B,D\right]+\left[A,C\right]BD+A\left[B,C\right]D \end{align*} \begin{align*} \left\{ AB,CD\right\} & =C\left\{ AB,D\right\} +\left[AB,C\right]D\cmt{\because\left\{ A,BC\right\} =B\left\{ A,C\right\} +\left[A,B\right]C}\\ & =C\left(A\left\{ B,D\right\} -\left[A,D\right]B\right)+\left(\left[A,C\right]B+A\left[B,C\right]\right)D\cmt{\because\left\{ AB,C\right\} =A\left\{ B,C\right\} -\left[A,C\right]B}\\ & =CA\left\{ B,D\right\} -C\left[A,D\right]B+\left[A,C\right]BD+A\left[B,C\right]D \end{align*} \begin{align*} \left\{ AB,CD\right\} & =\left\{ AB,C\right\} D-C\left[AB,D\right]\cmt{\because\left\{ A,BC\right\} =\left\{ A,B\right\} C-B\left[A,C\right]}\\ & =\left(\left\{ A,C\right\} B+A\left[B,C\right]\right)D-C\left(\left[A,D\right]B+A\left[B,D\right]\right)\cmt{\because\left\{ AB,C\right\} =\left\{ A,C\right\} B+A\left[B,C\right]}\\ & =\left\{ A,C\right\} BD+A\left[B,C\right]D-C\left[A,D\right]B-CA\left[B,D\right] \end{align*} \begin{align*} \left\{ AB,CD\right\} & =\left\{ AB,C\right\} D-C\left[AB,D\right]\cmt{\because\left\{ A,BC\right\} =\left\{ A,B\right\} C-B\left[A,C\right]}\\ & =\left(A\left\{ B,C\right\} -\left[A,C\right]B\right)D-C\left(\left[A,D\right]B+A\left[B,D\right]\right)\cmt{\because\left\{ A,BC\right\} =\left\{ A,B\right\} C-B\left[A,C\right]}\\ & =A\left\{ B,C\right\} D-\left[A,C\right]BD-C\left[A,D\right]B-CA\left[B,D\right] \end{align*}

(8)

\begin{align*} \left\{ A_{1}A_{2}\cdots A_{m},B_{1}B_{2}\cdots B_{n}\right\} & =\sum_{j\in\left\{ 1,2,\cdots,m\right\} }A_{1}\cdots A_{j-1}\left(\delta_{j,p}\left\{ A_{j},B_{1}B_{2}\cdots B_{n}\right\} +\sgn\left(j-p\right)\left[A_{j},B_{1}B_{2}\cdots B_{n}\right]\right)A_{j+1}\cdots A_{m}\\ & =\sum_{j\in\left\{ 1,2,\cdots,m\right\} }A_{1}\cdots A_{j-1}\left(\delta_{j,p}\left\{ B_{1}B_{2}\cdots B_{n},A_{j}\right\} +\sgn\left(j-p\right)\sum_{k\in\left\{ 1,2,\cdots,n\right\} }B_{1}\cdots B_{k-1}\left[A_{k},B_{k}\right]B_{k+1}\cdots B_{n}\right)A_{j+1}\cdots A_{m}\\ & =\sum_{j\in\left\{ 1,2,\cdots,m\right\} }A_{1}\cdots A_{k-1}\left(\delta_{j,p}\left(\sum_{k\in\left\{ 1,2,\cdots,n\right\} }B_{1}\cdots B_{k-1}\left(\delta_{k,q}\left\{ B_{k},A_{j}\right\} +\sgn\left(k-q\right)\left[B_{k},A_{j}\right]\right)B_{k+1}\cdots B_{n}\right)+\sgn\left(j-p\right)\sum_{k\in\left\{ 1,2,\cdots,n\right\} }B_{1}\cdots B_{k-1}\left[A_{j},B_{k}\right]B_{k+1}\cdots B_{n}\right)A_{j+1}\cdots A_{m}\\ & =\sum_{j\in\left\{ 1,2,\cdots,m\right\} }\sum_{k\in\left\{ 1,2,\cdots,n\right\} }A_{1}\cdots A_{j-1}B_{1}\cdots B_{k-1}\left(\delta_{j,p}\left(\delta_{k,q}\left\{ B_{k},A_{j}\right\} +\sgn\left(k-q\right)\left[B_{k},A_{j}\right]\right)+\sgn\left(j-p\right)\left[A_{j},B_{k}\right]\right)B_{k+1}\cdots B_{n}A_{j+1}\cdots A_{m}\\ & =\sum_{j\in\left\{ 1,2,\cdots,m\right\} }\sum_{k\in\left\{ 1,2,\cdots,n\right\} }A_{1}\cdots A_{j-1}B_{1}\cdots B_{k-1}\left(\delta_{j,p}\delta_{k,q}\left\{ A_{j},B_{k}\right\} -\delta_{j,p}\sgn\left(k-q\right)\left[A_{j},B_{k}\right]+\sgn\left(j-p\right)\left[A_{j},B_{k}\right]\right)B_{k+1}\cdots B_{n}A_{j+1}\cdots A_{m}\\ & =\sum_{j\in\left\{ 1,2,\cdots,m\right\} }\sum_{k\in\left\{ 1,2,\cdots,n\right\} }A_{1}\cdots A_{j-1}B_{1}\cdots B_{k-1}\left(\delta_{j,p}\delta_{k,q}\left\{ A_{j},B_{k}\right\} +\left(\sgn\left(j-p\right)-\delta_{j,p}\sgn\left(k-q\right)\right)\left[A_{j},B_{k}\right]\right)B_{k+1}\cdots B_{n}A_{j+1}\cdots A_{m} \end{align*} \(p=1,q=n\)とすると、
\[ \left\{ A_{1}A_{2}\cdots A_{m},B_{1}B_{2}\cdots B_{n}\right\} =B_{1}\cdots B_{n-1}\left\{ A_{1},B_{n}\right\} A_{2}\cdots A_{m}+\sum_{j\in\left\{ 1,\cdots,m\right\} ,k\in\left\{ 1,\cdots,n\right\} ,k\ne j}A_{1}\cdots A_{j-1}B_{1}\cdots B_{k-1}\left[A_{j},B_{k}\right]B_{k+1}\cdots B_{n}A_{j+1}\cdots A_{m} \] となり、\(p=m,q=1\)とすると、
\[ \left\{ A_{1}A_{2}\cdots A_{m},B_{1}B_{2}\cdots B_{n}\right\} =A_{1}\cdots A_{m-1}\left\{ A_{m},B_{1}\right\} B_{2}\cdots B_{n}-\sum_{j\in\left\{ 1,\cdots,m\right\} ,k\in\left\{ 1,\cdots,n\right\} ,k\ne j}A_{1}\cdots A_{j-1}B_{1}\cdots B_{k-1}\left[A_{j},B_{k}\right]B_{k+1}\cdots B_{n}A_{j+1}\cdots A_{m} \] となる。

(9)

\begin{align*} \left[AB,C\right] & =A\left[B,C\right]+\left[A,C\right]B\\ & =A\left(\left\{ B,C\right\} -2CB\right)+\left(\left\{ A,C\right\} -2CA\right)B\\ & =A\left\{ B,C\right\} +\left\{ A,C\right\} B-2\left(AC+CA\right)B\\ & =A\left\{ B,C\right\} +\left\{ A,C\right\} B-2\left\{ A,C\right\} B\\ & =A\left\{ B,C\right\} -\left\{ A,C\right\} B \end{align*}

(10)

(9)より、
\begin{align*} \left[A,BC\right] & =-\left[BC,A\right]\\ & =-\left(B\left\{ C,A\right\} -\left\{ B,A\right\} C\right)\\ & =\left\{ A,B\right\} C-B\left\{ A,C\right\} \end{align*} となるので与式は成り立つ。

(11)

\begin{align*} \left[\left\{ A,B\right\} ,C\right] & =\left[AB+BA,C\right]\\ & =\left[AB,C\right]+\left[BA,C\right]\\ & =ABC-CAB+BAC-CBA\\ & =ABC-\left(\left\{ C,A\right\} -AC\right)B+BAC-\left(\left\{ C,B\right\} -BC\right)A\\ & =ABC-\left\{ C,A\right\} B+ACB+BAC-\left\{ C,B\right\} A+BCA\\ & =A\left\{ B,C\right\} -\left\{ C,A\right\} B+B\left\{ A,C\right\} -\left\{ C,B\right\} A\\ & =-\left[\left\{ C,A\right\} ,B\right]-\left[\left\{ B,C\right\} ,A\right] \end{align*} これより、
\[ \left[\left\{ A,B\right\} ,C\right]+\left[\left\{ C,A\right\} ,B\right]+\left[\left\{ B,C\right\} ,A\right]=0 \] となるので与式は成り立つ。

(12)

\begin{align*} AB & =\frac{1}{2}\left(AB-BA+AB+BA\right)\\ & =\frac{1}{2}\left(\left[A,B\right]+\left\{ A,B\right\} \right) \end{align*}