Tensor algebra

1. $\underline{\boldsymbol{ v}}\boldsymbol{ a}\underline{\boldsymbol{ v}}$ becomes a scalar value, $x=\underline{\boldsymbol{ v}}\boldsymbol{ a}\underline{\boldsymbol{ v}}=v_i a_{ij} v_j$
2. $\boldsymbol{ x}\boldsymbol{ b}\underline{\boldsymbol{ u}}$ becomes a vector, $\underline{\boldsymbol{ v}}= \boldsymbol{ x}\boldsymbol{ b}\underline{\boldsymbol{ u}}$, $v_i=x_{ij}b_{jk}u_k$
3. $[\boldsymbol{ a}\otimes\boldsymbol{ a}]:[\underline{\boldsymbol{ v}}\otimes\underline{\boldsymbol{ v}}]$ is a 4th order tensor double contracted with a 2nd order tensor, which gives a 2nd order tensor, $\boldsymbol{ x}$. This becomes $x_{ij}= a_{ij} a_{kl} v_{k} v_{l}$

1. $x = \boldsymbol{ a}:\boldsymbol{ b}$
2. $\underline{\boldsymbol{ v}} = \boldsymbol{ a}\underline{\boldsymbol{ v}}$ (note order!)
3. $\textbf{\textsf{ A}} = \underline{\boldsymbol{ u}} \otimes \boldsymbol{ b} \otimes \underline{\boldsymbol{ v}}$

1. Short: Insert directly to index expressions, $\delta_{ij} a_{ik}a_{kj} - a_{kl}a_{kl} = a_{ik}a_{ki} - a_{kl}a_{kl}$. Then use that $a_{ik}=a_{ki}$, which gives $a_{ki}a_{ki}-a_{kl}a_{kl}=\boldsymbol{ a}:\boldsymbol{ a} - \boldsymbol{ a}:\boldsymbol{ a} = 0$
Longer: To get the index expression, we insert everything using basis vectors:
$[\delta_{ij} \underline{\boldsymbol{ e}}_i\otimes\underline{\boldsymbol{ e}}_j]:[[a_{kl} \underline{\boldsymbol{ e}}_k\otimes\underline{\boldsymbol{ e}}_l]\cdot[a_{mn} \underline{\boldsymbol{ e}}_m\otimes\underline{\boldsymbol{ e}}_{n}]] - [a_{op} \underline{\boldsymbol{ e}}_o\otimes\underline{\boldsymbol{ e}}_p] : [a_{qr} \underline{\boldsymbol{ e}}_q\otimes\underline{\boldsymbol{ e}}_r]$
Take all index coefficients outside
$\delta_{ij}a_{kl}a_{mn}[\underline{\boldsymbol{ e}}_i\otimes\underline{\boldsymbol{ e}}_j]:[[\underline{\boldsymbol{ e}}_k\otimes\underline{\boldsymbol{ e}}_l]\cdot[\underline{\boldsymbol{ e}}_m\otimes\underline{\boldsymbol{ e}}_{n}]] - a_{op}a_{qr}[\underline{\boldsymbol{ e}}_o\otimes\underline{\boldsymbol{ e}}_p]:[\underline{\boldsymbol{ e}}_q\otimes\underline{\boldsymbol{ e}}_r]$
Evaluate the single contraction (dot-product)
$\delta_{ij}a_{kl}a_{mn}[\underline{\boldsymbol{ e}}_i\otimes\underline{\boldsymbol{ e}}_j]:[\underline{\boldsymbol{ e}}_k\otimes\underline{\boldsymbol{ e}}_{n}\delta_{lm}] - a_{op}a_{qr}[\underline{\boldsymbol{ e}}_o\otimes\underline{\boldsymbol{ e}}_p]:[\underline{\boldsymbol{ e}}_q\otimes\underline{\boldsymbol{ e}}_r]$
Insert definition of the double contractions
$\delta_{ij}a_{kl}a_{mn}[\underline{\boldsymbol{ e}}_i\cdot \underline{\boldsymbol{ e}}_k][\underline{\boldsymbol{ e}}_j\cdot\underline{\boldsymbol{ e}}_{n}\delta_{lm}] - a_{op}a_{qr}[\underline{\boldsymbol{ e}}_o\cdot\underline{\boldsymbol{ e}}_q][\underline{\boldsymbol{ e}}_p\cdot\underline{\boldsymbol{ e}}_r]$
Evaluate the new single contractions
$\delta_{ij}a_{kl}a_{mn}[\delta_{ik}][\delta_{jn}\delta_{lm}] - a_{op}a_{qr}[\delta_{oq}][\delta_{pr}]$
Simplify all "$\delta_{ij}$" by exchanging replacing one index: "$x_{ij}\delta_{jk}=x_{ik}$"
$a_{jl}a_{lj} - a_{qp}a_{qp}$
And now we have a similar expression to the short route, and can finish following that one.
2. $z_{ij}\textsf{ Y}_{ ijkl}x_{kl} - [z_{ij} \textsf{ Y}_{ ijkl}]x_{kl} = 0$
3. $u_i x_{ij} v_j - [u_i v_j] x_{ij} = 0$

1. $a_{ij} \delta_{jk} = a_{ik}$, hence, $\boldsymbol{ a}\boldsymbol{ I}=\boldsymbol{ a}$. 2. $\textbf{\textsf{ X}}=\boldsymbol{ b}\overline{\otimes}\boldsymbol{ c}$ implies $\textsf{ X}_{ ijkl}=b_{ik}c_{jl}$, hence $[\delta_{ik} \delta_{jl}] a_{kl} = a_{ij}$ and $[\boldsymbol{ I}\overline{\otimes}\boldsymbol{ I}]:\boldsymbol{ a}=\boldsymbol{ a}$.
3. $\delta_{ij}[a_{ik} a_{kj}^{\mathrm{T}}] = \delta_{ij}[a_{ik} a_{jk}] = a_{ik} a_{ik} = \boldsymbol{ a}:\boldsymbol{ a}$

In all cases, we should show that rotating all tensors in the expressions by a proper orthogonal rotation tensor $\boldsymbol{ R}$ does not change the result initial expression. We will heavily use that $\boldsymbol{ R}^{\mathrm{T}}=\boldsymbol{ R}^{-1}$, such that $\boldsymbol{ R}^{\mathrm{T}}\boldsymbol{ R}=\boldsymbol{ I}$
1. $\underline{\boldsymbol{ v}}\cdot\underline{\boldsymbol{ u}}$
$[\boldsymbol{ R}\underline{\boldsymbol{ v}}]\cdot[\boldsymbol{ R}\underline{\boldsymbol{ u}}]=R_{ij}v_j R_{ik}u_k = R^{\mathrm{T}}_{ki}R_{ij} v_j u_k = R^{-1}_{ki}R_{ij} v_j u_k = \delta_{kj} v_j u_k = v_k u_k = \underline{\boldsymbol{ v}}\cdot\underline{\boldsymbol{ u}}$
2. $\boldsymbol{ a}:\boldsymbol{ a}$
$[\boldsymbol{ R}\boldsymbol{ a}\boldsymbol{ R}^{\mathrm{T}}]:[\boldsymbol{ R}\boldsymbol{ a}\boldsymbol{ R}^{\mathrm{T}}] = [R_{ij} a_{jk} R^{\mathrm{T}}_{kl}] [R_{im} a_{mn} R^{\mathrm{T}}_{nl}] = R^{\mathrm{T}}_{mi}R_{ij} R^{\mathrm{T}}_{kl} R_{ln} a_{jk} a_{mn} = \delta_{mj} \delta_{kn} a_{jk} a_{mn} = a_{mn} a_{mn} = \boldsymbol{ a}:\boldsymbol{ a}$
3. $\underline{\boldsymbol{ v}}\boldsymbol{ a}\underline{\boldsymbol{ v}}$
$[\boldsymbol{ R}\underline{\boldsymbol{ v}}][\boldsymbol{ R}\boldsymbol{ a}\boldsymbol{ R}^{\mathrm{T}}][\boldsymbol{ R}\underline{\boldsymbol{ v}}] = [R_{ij} v_j] [R_{ik} a_{kl} R^{\mathrm{T}}_{lm}][R_{mn} v_n] = R^{\mathrm{T}}_{ki} R_{ij} R^{\mathrm{T}}_{lm}R_{mn} v_j a_{kl} v_n = \delta_{kj} \delta_{ln} v_j a_{kl} v_n = v_k a_{kl} v_l = \underline{\boldsymbol{ v}}\boldsymbol{ a}\underline{\boldsymbol{ v}}$