Let Ω ⊂ ℝⁿ, n ≥ 2, be a bounded Lipschitz domain and 1 < q < ∞. We prove the inequality ∥T∥_Lq(Ω) ≤ C_DD (∥ dev T∥_Lq(Ω) + ∥ Div T∥_Lq(Ω)) being valid for tensor fields T : Ω → ℝ^nxn with a normal boundary condition on some open and non-empty part Γ_ν of the boundary ∂Ω. Here dev T = T - 1/n tr (T) · Id denotes the deviatoric part of the tensor T and Div is the divergence row-wise. Furthermore, we prove ∥T∥_L2(Ω) ≤ C_DSC (∥ dev sym T∥_L2(Ω) + ∥ Curl T∥_L2(Ω)) if n ≥ 3, ∥T∥_L2(Ω) ≤ C_DSDC (∥ dev sym T∥_L2(Ω) + ∥ dev Curl T∥_L2(Ω)) if n = 3, being valid for tensor fields T with a tangential boundary condition on some open and non-empty part Γ_τ of ∂Ω. Here, sym T = 1/2 (T + T^T) denotes the symmetric part of T and Curl is the rotation row-wise.